Semi-classical Analysis Around Local Maxima and Saddle Points for Degenerate Nonlinear Choquard Equations

Abstract

We study existence of semi-classical states for the nonlinear Choquard equation:

$$\begin{aligned} -\varepsilon ^2\Delta v+ V(x)v = {1\over \varepsilon ^\alpha }(I_\alpha *F(v))f(v) \quad \text {in}\ {\mathbb {R}}^N, \end{aligned}$$

where $N\ge 3$, $\alpha \in (0,N)$, $I_\alpha (x)=A_\alpha /|{x}|^{N-\alpha }$ is the Riesz potential, $F\in C^1({\mathbb {R}},{\mathbb {R}})$, $F'(s)=f(s)$ and $\varepsilon >0$ is a small parameter. We develop a new variational approach, in which our deformation flow is generated through a flow in an augmented space to get a suitable compactness property and to reflect the properties of the potential. Furthermore our flow keeps the size of the tails of the function small and it enables us to find a critical point without introducing a penalization term. We show the existence of a family of solutions concentrating to a local maximum or a saddle point of $V(x)\in C^N({\mathbb {R}}^N,{\mathbb {R}})$ under general conditions on F(s). Our results extend the results by Moroz and Van Schaftingen (Calc Var Partial Differ Equ 52:199–235, 2015) for local minima (see also Cingolani and Tanaka (Rev Mat Iberoam 35(6):1885–1924, 2019)) and Wei and Winter (J Math Phys 50:012905, 2009) for non-degenerate critical points of the potential.

1 Introduction

In the recent years a large amount of papers has been devoted to investigate concentration phenomena of solutions to nonlinear Schrödinger equations with local sources around potential wells, namely local minima of some external potential functions. Starting to the celebrated papers by Floer and Weinstein [31] and Rabinowitz [58], several variational approaches were implemented and some efforts were done to obtain optimal results. We mention for instance [7, 16, 18,19,20,21, 29, 36, 37]. A more difficult problem seems to detect concentration phenomena around local maxima or saddle points of the potential type function. Some results are known for nonlinear Schrödinger equations under nondegeneracy conditions of the local maxima which allow to perform Lyapunov Schmidt reduction arguments [2, 3, 31, 41, 51]. More recently, del Pino and Felmer in [30] introduced a new reduction and proved a concentration result for solutions of nonlinear Schrödinger equation around local maxima and saddle points of the potential, assuming Ambrosetti-Rabinowitz type conditions and monotonicity conditions on the nonlinearity, which are crucial to apply a Nehari manifold approach. We refer to [28] for a generalization of the result of [30]. The more general result is contained in [8, 9] where Byeon and the second author succeeded to show the existence of families of solutions to nonlinear Schrödinger equations with local nonlinearity of Berestycki-Lions type concentrating at critical points which are given by minimax method with suitable linking properties, e.g. local maxima, mountain pass critical points, non-degenerate critical points. See also [6, 10,11,12, 39].

The goal of the present paper is to develop a new theoretical approach to obtain existence of solutions which concentrate at local maxima or saddle points of potential functions, under quite optimal assumptions on the nonlinearity and without any nondegeneracy conditions for class of nonlinear Schrödinger equations having local or nonlocal source.

As prototype of nonlocal problem in the source, we focus our analysis on the following class of equations

$$\begin{aligned} \left\{ \begin{aligned} -&\varepsilon ^2\Delta v+V(x)v = {1\over \varepsilon ^\alpha }(I_\alpha *F(v)) f(v) \quad \text {in}\ {\mathbb {R}}^N, \\&v>0 \quad \text {in}\ {\mathbb {R}}^N, \quad v\in H^1({\mathbb {R}}^N), \end{aligned} \right. \end{aligned}$$

(1.1)

where $\varepsilon >0$ is a small positive parameter, $N\ge 3$, $\alpha \in (0,N)$,

$$\begin{aligned} I_\alpha (x) = {\Gamma ({N-\alpha \over 2})\over \Gamma ({\alpha \over 2})\pi ^\alpha |{x}|^{N-\alpha }}:{\mathbb {R}}^N\setminus \{0\}\rightarrow {\mathbb {R}}\end{aligned}$$

is the Riesz potential, $F(s)\in C^1({\mathbb {R}},{\mathbb {R}})$ and $f(s)=F'(s)$. We recall that in 1954 the Eq. (1.1) with $N=3$, $\alpha =2$ and $F(s)={1 \over 2}|{s}|^2$ was introduced by Pekar [52] to describe the quantum theory of a polaron at rest. In 1976, (1.1) appeared in the work of Choquard on the modeling of an electron trapped in its own hole, in a certain approximation to the Hartree-Fock theory of plasma (see also [32]). More recently it has found a great attention due to models of self-gravitational collapse of a quantum mechanical wave function, proposed by Roger Penrose [53,54,55] and in that context it is known as as Schrödinger-Newton equation (see also [46, 60]).

In literature, (1.1) is usually referred as nonlinear Choquard equation or Schrödinger equation with Hartree type potential. From a mathematical point of view, the early existence and symmetry results are due to Lieb [42] and Lions [43]. Successively, Ma and Zhao [44] classified all positive solutions to (1.1) for power nonlinearity and showed that they must be radially symmetric and monotonically decreasing about some fixed point. Recently Moroz and Van Schaftingen [48] investigated existence, some qualitative properties and decay asymptotics of positive ground state solutions to (1.1) for $\varepsilon >0$ fixed when F satisfies the Berestycki-Lions type conditions. Other results are contained in [4, 13, 17, 18, 24, 27, 40, 47, 50, 57].

In the present paper we are interested in the study the existence of concentrating family of solutions of (1.1) at local maxima or saddle point of V(x) as $\varepsilon \rightarrow 0$.

Denoting $u(x)=v(\varepsilon x)$, the Eq. (1.1) is equivalent to

$$\begin{aligned} \left\{ \begin{aligned} -&\Delta u+V(\varepsilon x)u = (I_\alpha *F(u)) f(u) \quad \text {in}\ {\mathbb {R}}^N, \\&u>0 \quad \text {in}\ {\mathbb {R}}^N. \quad u\in H^1({\mathbb {R}}^N), \end{aligned} \right. \end{aligned}$$

(1.2)

Thus we try to find critical points of the corresponding functional:

$$\begin{aligned} I_\varepsilon (u)={1 \over 2}\int _{{\mathbb {R}}^N}|{\nabla u}|^2+V(\varepsilon x)u^2 -{1 \over 2}\int _{{\mathbb {R}}^N}(I_\alpha *F(u))F(u):H^1({\mathbb {R}}^N)\rightarrow {\mathbb {R}}\end{aligned}$$

and we ask the existence of a concentrating family $(u_\varepsilon )$ of solutions of (1.2) as $\varepsilon \rightarrow 0$.

Firstly the concentration at nondegenerate critical points of the potential V(x) has been studied by Wei and Winter [62] using Lyapunov Schmidt reduction when $N=3$, $\alpha =2$ and $F(s)= s^2$. The case of local minima (possibly degenerate) of V when $N=3$ and $F(s)=s^2$ has been considered in [22] by means of a penalization approach (see also [14, 59, 63]). More recently, Moroz and Van Schaftingen [49] proved existence of a single-peak solution of (1.1) concentrating at a local minima of V(x) for $f(s)=|{s}|^{p-2}s$, $p\in [2,{N+\alpha \over N-2})$ via a new non-local penalization method. [64] extended the result in [49] and showed the existence under (f4) below, $\lim _{t\rightarrow \infty } {f(t)\over t^{\alpha +2\over N-2}}=0$ and

$$\begin{aligned} \lim _{t\rightarrow 0}{f(t)\over t}=0. \end{aligned}$$

(1.3)

They also proved the existence of multi-peak solutions, whose each peak concentrates at different local minimum of V(x) as $\varepsilon \rightarrow 0$. We note that conditions $p\ge 2$ or (1.3) is important in their arguments as it enables them to use linearized problems at infinity. See also [1, 45, 56] dealing with critical Choquard equations.

In [23] we developed a new variational approach which is applicable to a wide class of nonlinearities including $F(s)=|{s}|^p$, $p\in ({N+\alpha \over N}, {N+\alpha \over N-2})$. In particular, we can deal with the sublinear case $p\in ({N+\alpha \over N}, 2)$, differently to [49]. We obtained the multiplicity of concentrating solutions via the cup-length of a critical set $\textrm{Crit}_{V_0}=\{ x\in \Omega ;\, V(x)=V_0\}$, where $\Omega \subset {\mathbb {R}}^N$ is a bounded set such that $V_0\equiv \inf _{x\in \Omega } V(x) < \inf _{x\in \partial \Omega } V(x)$. See also [38] for the effect of the topology of the potential wells on the existence of multi-bumps solutions.

The main purpose of this paper is to study the existence of concentrating family of solutions of nonlinear Choquard equation (1.1) at a local maximum or saddle point of V(x). To our knowledge, the only concentration result dealing nondegenerate local maxima is due to Wei and Winter [62], when $N=3$, $\alpha =2$ and $F(s)=s^2$.

The existence of concentrating families of solutions at local maxima and saddle points of V(x) is a more involved open problem and deformation argument using the standard gradient flow associated to $I_\varepsilon (u)$ does not seem enough. We also note that non-degeneracy of solutions of the limit problem $-\Delta u + V(x_0)u = (I_\alpha *F(u))f(u)$ is not known except the case $N=3$, $\alpha =2$, $F(u)=|{u}|^2$ and it seems difficult to apply Lyapunov Schmidt reduction methods in general.

To show the existence of concentrating family of solutions, in this paper we develop a new deformation argument, which is partially inspired by [8, 25, 33, 35].

Our deformation argument is developed for $V(x)\in C^1({\mathbb {R}}^N,{\mathbb {R}})$ through a deformation in an augmented space ${\mathbb {R}}^N\times H^1({\mathbb {R}}^N)$ and it has the following new features:

• (i) Our deformation flow is developed through a deformation for an augmented functional:

  $$\begin{aligned} J_\varepsilon (z,u) ={1 \over 2}\int _{{\mathbb {R}}^N}|{\nabla u}|^2 + {1 \over 2}\int _{{\mathbb {R}}^N}V(\varepsilon x+z)u(x)^2 - {1 \over 2}\int _{{\mathbb {R}}^N}(I_\alpha *F(u))F(u) \end{aligned}$$

  for all $(z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$. We use the following translation of $u\in H^1({\mathbb {R}}^N)$ as a part of our new deformation argument:

  $$\begin{aligned} t\mapsto u\left( x-{h\over \varepsilon }t\right) ;\, (-\delta ,\delta )\rightarrow {\mathbb {R}}, \end{aligned}$$

  (1.4)

  where $h\in {\mathbb {R}}^N$. If $u_\varepsilon (x)$ “concentrates” at some point $p_0\in {\mathbb {R}}^N$ in the original scale for (1.1), that is, $u_\varepsilon (x)\sim v(x-{p_0\over \varepsilon })$ for some function v(x), then as $\varepsilon \sim 0$

  $$\begin{aligned} \begin{aligned}&{d\over dt}\Bigr |_{t=0} I_\varepsilon \left( u_\varepsilon \left( x-{h\over \varepsilon }t\right) \right) = {1 \over 2}{d\over dt}\Bigr |_{t=0} \int _{{\mathbb {R}}^N}V(\varepsilon x)u_\varepsilon \left( x-{h\over \varepsilon }t\right) ^2\, dx \\&\quad \quad \sim {1 \over 2}{d\over dt}\Bigr |_{t=0} \int _{{\mathbb {R}}^N}V(\varepsilon x)v\left( x-{p_0+ht\over \varepsilon }\right) ^2\, dx\\&\quad \quad = {1 \over 2}{d\over dt}\Bigr |_{t=0} \int _{{\mathbb {R}}^N}V(\varepsilon x+p_0+ht)v(x)^2\, dx \\&\quad \quad = {1 \over 2}(\nabla V(p_0),h)\int _{{\mathbb {R}}^N}v(x)^2\,dx. \end{aligned} \end{aligned}$$

  Thus, if $\nabla V(p_0)\not =0$, choosing $h=\nabla V(p_0)$, the traslation flow (1.4) gives a decreasing flow for $I_\varepsilon (u)$ in a small neighborhood of $u_\varepsilon$. Thus $\nabla V(p_0)$ gives a useful information for deformation argument. However we note that in $H^1({\mathbb {R}}^N)$ the flow (1.4) is continuous but not of class $C^1$ in general and it cannot be obtained through the standard deformation theory, where the flow is obtained as a solution of ODE in a Banach space and it must be of class $C^1$.

  Our augmented functional $J_\varepsilon (z,u)$ enjoys the following property:

  $$\begin{aligned} J_\varepsilon (z,u)=I_\varepsilon \left( u\left( x-{z\over \varepsilon }\right) \right) \quad \text {for all}\ z\in {\mathbb {R}}^N \ \text {and}\ u\in {\mathbb {R}}\end{aligned}$$

  and the traslation flow (1.4) can be obtained as a composition of a $C^1$-flow in the augmented space

  $$\begin{aligned} t\mapsto (ht,u(x));\, (-\delta ,\delta )\rightarrow {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}\end{aligned}$$

  and a projection

  $$\begin{aligned} \pi _\varepsilon :\, (z,u)\mapsto u\left( x-{z\over \varepsilon }\right) ;\, {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}\rightarrow H^1({\mathbb {R}}^N). \end{aligned}$$

  We also note that the standard deformation flow $\eta (t):\, (-\delta ,\delta )\rightarrow H^1({\mathbb {R}}^N)$ for $I_\varepsilon (u)$ in $H^1({\mathbb {R}}^N)$ also can be obtained as a composition of a flow $(-\delta ,\delta )\rightarrow {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)};\, t\mapsto (0,\eta (t))$ and the projection $\pi _\varepsilon$. In the following sections, first we construct a deformation flow ${\widetilde{\eta }}$ for the augmented functional $J_\varepsilon (z,u)$ in ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ and we construct a deformation flow for $I_\varepsilon (u)$ as a composition $(\pi _\varepsilon \circ {\widetilde{\eta }})(t)$. We also note that our new construction of a deformation flow works under weaker version of Palais-Smale type condition (see Proposition 4.5, 4.7 and 6.1).

• (ii) Another new aspect of our deformation flow is that it keeps the size of the tail of functions small during deformation. That is, defining the size of a tail of a function u by

  $$\begin{aligned} T_\varepsilon (u)=\int _{{\mathbb {R}}^N}{\widetilde{\zeta }}_{4/\sqrt{\varepsilon }}(x-\beta (u))(|{\nabla u}|^2+u^2)\,dx, \end{aligned}$$

  where ${\widetilde{\zeta }}_R(x)\in C^\infty ({\mathbb {R}}^N,{\mathbb {R}})$ satisfies ${\widetilde{\zeta }}_R(x)=1$ for $|{x}|\ge R$ and ${\widetilde{\zeta }}_R(x)=0$ for $|{x}|\le R-1$ and $\beta (u)$ is the “center of mass” of u which will be defined in Sect. 3.3. We observe that for small $\kappa _\varepsilon$ with $\kappa _\varepsilon \rightarrow 0$, the set $\{u:\, T_\varepsilon (u)\le \kappa _\varepsilon \}$ is positively invariant under our deformation flow. See Proposition 6.1 and (6.3) in Sect. 6. This property ensures that if u(x) concentrates around the center $\beta (u)$ of mass, deformed function $\eta (t,u)$ continues to concentrate around the center $\beta (\eta (t,u))$ of mass of the deformed functions $\eta (t,u)$. The standard deformation flow does not have this property. Such a property is usually obtained by using tail minimization methods for local problems, that is, we solve the elliptic boundary problem outside of a large ball centered at $\beta (u)$. We note that such a tail minimizing problem requires the unique solvability of the elliptic boundary problem and usually it is ensured for local problems, i.e., for nonlinear Schrödinger equations, under the condition $f\in C^1$. For non-local problems, e.g. nonlinear Choquard equations such an approach does not work because of non-local feature of the problem. In Sects. 5 and 6 we develop a new deformation method in which the deformation flow is constructed through a deformation in an augmented space ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$. Our deformation method works for both of local and non-local problems. In a paper in preparation, we aim to apply this new approach to fractional problem (see [15] for concentration around local minima). See Remark 8.3 in Sect. 8 for an application to local problem (see also [26]).

Remark 1.1

In [8, 9], a related deformation argument is developed for nonlinear Schrödinger equation:

$$\begin{aligned} -\Delta u+V(\varepsilon x)u = g(u) \quad \text {in}\ {\mathbb {R}}^N \end{aligned}$$

(1.5)

in a different way. Namely it is constructed as an iteration of 3 flows:

• (1) The standard deformation flow $\eta _1(t,\cdot )$. Here $\eta _1(t,\cdot )$ is a solution of ${d\eta _1\over dt} = -\varphi (\eta _1)\mathcal{V}(\eta _1)$, $\eta _1(0,u)=u$, where $\mathcal{V}(\cdot )$ is a pseudo-gradient vector associated to the functional corresponding to (1.2).

• (2) The translation flow $\eta _2(t,u)(x)=u(x-{h\over \varepsilon }t)$. Here $h=-\nabla V(\varepsilon \beta (u))$, where $\beta (u)$ is the center of mass of u.

• (3) The tail minimizing operator $\tau _\varepsilon (u)$, which is defined by $\tau _\varepsilon (u)=v$, where v is a solution of the exterior problem:

  $$\begin{aligned} \left\{ \begin{aligned} -&\Delta v+V(\varepsilon x)v = g(v) \quad \text {in}\ |{x-\beta (u)}| >R, \\&v(x)=u(x) \quad \text {on}\ |{x-\beta (u)}|=R. \end{aligned} \right. \end{aligned}$$

  (1.6)

The procedure is rather complicated and in present paper we give an “easier” deformation argument through a construction flow in an augmented space ${\mathbb {R}}^N\times H^1({\mathbb {R}}^N)$. We note that the exterior problem (1.6) is well-defined for local problem (1.5). But for non-local problem (1.2), the exterior problem is not well-defined because of non-locality of the problem.

To state our existence result for (1.2), we assume

• (f1) $f(s)\in C({\mathbb {R}},{\mathbb {R}})$;

• (f2) there exists $C>0$ such that for all $s\in {\mathbb {R}}$

  $$\begin{aligned} |{sf(s)}| \le C\left( |{s}|^{N+\alpha \over N} +|{s}|^{N+\alpha \over N-2}\right) ; \end{aligned}$$

• (f3) $F(s)=\int _0^s f(t)\, dt$ satisfies

  $$\begin{aligned} \lim _{s\rightarrow 0} {F(s)\over |{s}|^{N+\alpha \over N}}=0, \quad \lim _{s\rightarrow \infty } {F(s)\over |{s}|^{N+\alpha \over N-2}}=0; \end{aligned}$$

• (f4) f(s) is odd and f is positive on $(0,\infty )$.

We remark that the conditions (f1)–(f4) are in the spirit of Berestycki and Lions [5, 34, 48] and in our previous work [23] for a continuous potential V(x) we studied concentration at a local minimum under these conditions.

In the present paper we require much regularity on the potential V(x). Precisely for V(x) we assume

• (V1) $V(x)\in C^N({\mathbb {R}}^N,{\mathbb {R}})$, $\nabla V(x)\in L^{N\over 2}({\mathbb {R}}^N) + L^\infty ({\mathbb {R}}^N)$;

• (V2) $\inf _{x\in {\mathbb {R}}^N} V(x) \equiv {\underline{V}} >0$, $\sup _{x\in {\mathbb {R}}^N} V(x) \equiv {\overline{V}}<\infty$;

• (V3) there exists a bounded connected open set $\Omega \subset {\mathbb {R}}^N$ with a smooth boundary $\partial \Omega$ such that

  $$\begin{aligned} \nabla V(x)\not = 0 \qquad \text {for all}\ x\in \partial \Omega . \end{aligned}$$

We mainly study two situations where V(x) has a local maximum in $\Omega$ or V(x) has a mountain pass geometry in $\Omega$. More precisely, we assume (LM) or (MP) below.

• (LM) $V_0\equiv \sup _{x\in \Omega } V(x) > \sup _{x\in \partial \Omega } V(x)$;

• (MP) There exist $e_0$, $e_1\in \Omega$ such that setting

  $$\begin{aligned} \begin{aligned}&V_0 \equiv \inf _{c\in \Lambda } \max _{\xi \in [0,1]} V(c(\xi )), \\&\Lambda =\{c(\xi )\in C([0,1],\Omega ):\, c(0)=e_0,\, c(1)=e_1\}, \end{aligned} \end{aligned}$$

  $V_0$ satisfies

  • (i) $V(e_0)$, $V(e_1)<V_0$;

  • (ii) for $x\in \partial \Omega$ with $V(x)=V_0$,

    $$\begin{aligned} -\nabla V(x) \not \in \{\mu n(x):\, \mu \ge 0\}, \end{aligned}$$

    where $n(x)\in {\mathbb {R}}^N$ is the unit outer normal at $x\in \partial \Omega$.

We note that under the assumption (i), (ii) it is standard to see that $V_0$ is a critical value of V(x).

Our main result is

Theorem 1.2

Assume (f1)–(f4) and (V1)–(V3). Moreover suppose (LM) or (MP). Then (1.1) has at least one positive solution concentrating in

$$\begin{aligned} \textrm{Crit}_{V_0} \equiv \{ x\in \Omega :\, V(x)=V_0,\, \nabla V(x)=0\}. \end{aligned}$$

That is, there exist $\varepsilon _0>0$ and a family $(u_\varepsilon )_{\varepsilon \in (0,\varepsilon _0]}$ of solutions of (1.2) with the following property: for any sequence $(\varepsilon _j)_{j=1}^\infty \subset (0,\varepsilon _0]$ with $\varepsilon _j\rightarrow 0$ after extracting a subsequence—we denote it by $\varepsilon _j$ for simplicity of notation—, there exist $(x_j)_{j=1}^\infty \subset {\mathbb {R}}^N$, $x_0\in \textrm{Crit}_{V_0}$ and a least energy solution $\omega _0\in H^1({\mathbb {R}}^N)$ of the limit problem $-\Delta u + V(x_0)u = (I_\alpha *F(u))f(u)$ in ${\mathbb {R}}^N$ such that

$$\begin{aligned} \begin{aligned}&\varepsilon _j x_j\rightarrow x_0, \\&u_{\varepsilon _j}(x-x_j)\rightarrow \omega _0(x) \quad \text {strongly in}\ H^1({\mathbb {R}}^N)\quad \text {as}\ j\rightarrow \infty . \end{aligned} \end{aligned}$$

In (V1)–(V3), the assumption $V(x)\in C^N({\mathbb {R}}^N,{\mathbb {R}})$ is used in order to show via Sard’s Theorem that the set of critical values of V(x) is of measure 0. For a potential V(x) of class $C^1$, we can show the existence of a solution under the following assumption of isolatedness of critical points of V(x)

• (V1’) $V(x)\in C^1({\mathbb {R}}^N,{\mathbb {R}})$, $\nabla V(x)\in L^{N\over 2}({\mathbb {R}}^N)+L^\infty ({\mathbb {R}}^N)$;

• (V1”) critical points of V(x) in $\Omega$ are isolated in $\Omega$.

Namely we have

Theorem 1.3

Assume (f1)–(f4) and (V1’), (V1”), (V2), (V3). Moreover suppose (LM) or (MP). Then the conclusion of Theorem 1.2 holds.

Remark 1.4

If we assume (V1’) without (V1”) instead of (V1) in Theorem 1.2, a weaker version of the result holds. See Sect. 7.4.

This paper is organized as follows: In Sect. 2 we give some preliminary results. In Sect. 3 we study the limit problem. We introduce a Pohozaev type function $P_a(u)$ and a center $\beta (u)$ of mass, which are used in this paper repeatedly. In Sect. 4 we introduce a neighborhood of expected solutions and we show a concentration-compactness type results for functional $I_\varepsilon (u)$. We will develop a local deformation argument in this neighborhood in Sects. 5, 6, and 7. Here newly introduced $\varepsilon$-dependent distance $\mathop {\textrm{dist}}\nolimits _\varepsilon (\cdot ,\cdot )$ in $H^1({\mathbb {R}}^N)$ plays an important role. In Sect. 5 we introduce a functional $T_\varepsilon (u)$ to estimate the size of the tail of functions u and we construct a vector field, which decreases both of $T_\varepsilon (u)$ and $I_\varepsilon (u)$ and which enables us to generate a special deformation flow that keeps the tail of functions small. In Sect. 6 we give our new deformation result for $I_\varepsilon (u)$, which has new features stated above. Finally we give a proof of our main existence result in Sect. 7. In Sect. 8 we give a remark on concentration at a local minimum of V(x).

2 Preliminaries

In what follows, we use notation: for $u\in H^1({\mathbb {R}}^N)$

$$\begin{aligned} \begin{aligned}&\Vert {u}\Vert _{H^1} = \left( \int _{{\mathbb {R}}^N}|{\nabla u}|^2+u^2\right) ^{1/2}, \\&\Vert {u}\Vert _r = \left( \int _{{\mathbb {R}}^N}|{u}|^r\right) ^{1/r} \quad \text {for}\ r\in [1,\infty ), \qquad \Vert {u}\Vert _\infty = \mathop {\mathrm{ess\, sup}}_{x\in {\mathbb {R}}^N} |{u(x)}|. \end{aligned} \end{aligned}$$

We also use notation for $p\in {\mathbb {R}}^N$, $u_0\in H^1({\mathbb {R}}^N)$, $r>0$

$$\begin{aligned} \begin{aligned}&B(p,r) = \{ x\in {\mathbb {R}}^N:\, |{x-p}|<r \}, \qquad {\overline{B}}(p,r) = \{ x\in {\mathbb {R}}^N:\, |{x-p}|\le r \}, \\&B_{H^1}(u_0,r) = \{ u\in H^1({\mathbb {R}}^N):\, \Vert {u-u_0}\Vert _{H^1} <r\}. \end{aligned} \end{aligned}$$

2.1 Estimates for Non-local Term

First we give some estimates for $\int _{{\mathbb {R}}^N}(I_\alpha *f)g$ and

$$\begin{aligned} \mathcal{D}(u) = \int _{{\mathbb {R}}^N}(I_\alpha *F(u))F(u). \end{aligned}$$

For proofs, we refer to [23].

We denote various constants, which are independent of u, by C, $C'$, $C''$, $\cdots$

Lemma 2.1

(c.f. Section 2.1 of [23]).

• (i) For p, $r>1$ and $\alpha \in (0,N)$ with ${1\over p}+{1\over r}= {N+\alpha \over N}$ there exists a constant $C=C(N,\alpha ,p,r)>0$ such that

  $$\begin{aligned} \left| {\int _{{\mathbb {R}}^N}(I_\alpha *f)g}\right| \le C\Vert {f}\Vert _p \Vert {g}\Vert _r \end{aligned}$$

  for all $f\in L^p({\mathbb {R}}^N)$, $g\in L^r({\mathbb {R}}^N)$.

• (ii) Assume p, $r>1$ and $\alpha \in (0,N)$ with ${1\over p}+{1\over r}< {N+\alpha \over N}$. Then for $L\ge 1$ there exists a constant $D_L=D_L(N,\alpha ,p,r)>0$ such that $D_L \rightarrow 0$ as $L\rightarrow \infty$ and

  $$\begin{aligned} \left| {\int _{{\mathbb {R}}^N}(I_\alpha *f)g}\right| \le D_L \Vert {f}\Vert _p \Vert {g}\Vert _r \end{aligned}$$

  for all $f\in L^p({\mathbb {R}}^N)$, $g\in L^r({\mathbb {R}}^N)$ with $\mathop {\textrm{dist}}\nolimits (\mathop {\textrm{supp}}f, \mathop {\textrm{supp}}g)\ge L$. $\square$

In (ii), $D_L$ is given by

$$\begin{aligned} D_L = \Vert {I_\alpha ^L}\Vert _q, \end{aligned}$$

where q satisfies ${1\over p}+{1\over q} + {1\over r}=2$, in particular $q>{N\over N-\alpha }$ and $I_\alpha ^L(x)$ is defined by

$$\begin{aligned} I_\alpha ^L(x)={\left\{ \begin{array}{ll} {1\over |{x}|^{N-\alpha }} &{}\text {for}\ |{x}|\ge L,\\ 0 &{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

Setting $\sigma (s)=s^2 +|{s}|^{2N\over N-2}$ for $s\in {\mathbb {R}}$, under (f2) we have for u, $v\in H^1({\mathbb {R}}^N)$

$$\begin{aligned} \begin{aligned}&\Vert {F(u)}\Vert _{2N\over N+\alpha } \le C\sigma (\Vert {u}\Vert _{H^1})^{N+\alpha \over 2N}, \\&|{\mathcal{D}(u)}| \le C\Vert {F(u)}\Vert _{2N\over N+\alpha }^2 \le C'\sigma (\Vert {u}\Vert _{H^1})^{N+\alpha \over N}, \\&|{\mathcal{D}'(u)v}| \le C\Vert {F(u)}\Vert _{2N\over N+\alpha }\Vert {f(u)v}\Vert _{2N\over N+\alpha } \le C'\sigma (\Vert {u}\Vert _{H^1})^{N+\alpha \over 2N} (\Vert {u}\Vert _{H^1}^{\alpha \over N}+\Vert {u}\Vert _{H^1}^{\alpha +2\over N-2})\Vert {v}\Vert _{H^1}. \end{aligned} \end{aligned}$$

We also have

$$\begin{aligned} \begin{aligned} I_\varepsilon (u)&\ge {1 \over 2}\Vert {\nabla u}\Vert _2^2 +{1 \over 2}{\underline{V}}\Vert {u}\Vert _2^2 - C'\sigma (\Vert {u}\Vert _{H^1})^{N+\alpha \over N}, \\ I_\varepsilon '(u)u&\ge \Vert {\nabla u}\Vert _2^2 + {\underline{V}}\Vert {u}\Vert _2^2 - C'\sigma (\Vert {u}\Vert _{H^1})^{N+\alpha \over N}. \end{aligned} \end{aligned}$$

In particular, $I_\varepsilon (u)$ has mountain pass geometry uniformly in $\varepsilon \in (0,1]$ and we have

Corollary 2.2

There exist $\rho _0>0$ and $c_0>0$ such that for $\varepsilon \in (0,1]$

$$\begin{aligned} I_\varepsilon (u) \ge c_0\Vert {u}\Vert _{H^1}^2, \quad I_\varepsilon '(u)u \ge c_0\Vert {u}\Vert _{H^1}^2 \end{aligned}$$

for all $u\in H^1({\mathbb {R}}^N)$ with $\Vert {u}\Vert _{H^1} \le \rho _0$. $\square$

For $R>0$ we choose functions $\zeta _R(s)$, ${\widetilde{\zeta }}_R(s)\in C^\infty ({\mathbb {R}}^N,{\mathbb {R}})$ such that

$$\begin{aligned}&\zeta _R(x)= {\left\{ \begin{array}{ll} 1 &{}\text {for}\ |{x}|\le R,\\ 0 &{}\text {for}\ |{x}|\ge R+1, \end{array}\right. } \quad {\widetilde{\zeta }}_R(x) = {\left\{ \begin{array}{ll} 0 &{}\text {for}\ |{x}|\le R-1,\\ 1 &{}\text {for}\ |{x}|\ge R, \end{array}\right. } \nonumber \\&\zeta _R(x),\, {\widetilde{\zeta }}_R(x)\in [0,1],\ |{\nabla \zeta _R(x)}|,\, |{\nabla {\widetilde{\zeta }}_R(x)}|\le 2 \quad \text {for all}\ x\in {\mathbb {R}}^N. \end{aligned}$$

(2.1)

We will use the following inequalities frequently: for $u\in H^1({\mathbb {R}}^N)$, $R>0$, $p\in {\mathbb {R}}^N$

$$\begin{aligned} \Vert {\zeta _R(x-p)u}\Vert _{H^1} \le 3\Vert {u}\Vert _{H^1}, \quad \Vert {{\widetilde{\zeta }}_R(x-p)u}\Vert _{H^1} \le 3\Vert {u}\Vert _{H^1}. \end{aligned}$$

(2.2)

In fact,

$$\begin{aligned} \begin{aligned} \Vert {\zeta _R(x-p)u}\Vert _{H^1}^2&= \Vert {\nabla (\zeta _R(x-p)u)}\Vert _2^2 + \Vert {\zeta _R(x-p)u}\Vert _2^2 \\&\le 2\Vert {\zeta _R(x-p)\nabla u}\Vert _2^2 +2\Vert {(\nabla \zeta _R(x-p))u)}\Vert _2^2 + \Vert {u}\Vert _2^2 \\&\le 2\Vert {\nabla u}\Vert _2^2 + 9\Vert {u}\Vert _2^2 \le 9\Vert {u}\Vert _{H^1}^2. \end{aligned} \end{aligned}$$

We can show the second inequality in a similar way.

Lemma 2.3

(c.f. Corollary 2.6 of [23]). For a fixed $M>0$ there exists $C>0$ such that for any R, $L\ge 1$ and $u\in H^1({\mathbb {R}}^N)$ with $\Vert {u}\Vert _{H^1}\le M$

• (i) $|{(\mathcal{D}'(u)-\mathcal{D}'(\zeta _R u))\zeta _R u}| \le C(D_L+\sigma (\Vert {u}\Vert _{H^1(|{x}|\in [R,R+L])})^{N+\alpha \over 2N})$.

• (ii) $|{(\mathcal{D}'(u)-\mathcal{D}'({\widetilde{\zeta }}_{R+L} u))\zeta _{R+L} u}| \le C(D_L+\sigma (\Vert {u}\Vert _{H^1(|{x}|\in [R,R+L])})^{N+\alpha \over 2N})$.

Here $D_L>0$ is given in Lemma 2.1. In particular $D_L\rightarrow 0$ as $L\rightarrow \infty$.

Proof

We set

$$\begin{aligned} \chi _1(x)= & {} {\left\{ \begin{array}{ll} 1 &{}\text {if}\ |{x}|\le R,\\ 0 &{}\text {otherwise,} \end{array}\right. }\ \chi _2(x)={\left\{ \begin{array}{ll} 1 &{}\text {if}\ |{x}|\in [R,R+L],\\ 0 &{}\text {otherwise,} \end{array}\right. }\\ \chi _3(x)= & {} {\left\{ \begin{array}{ll} 1 &{}\text {if}\ |{x}|\ge R+L,\\ 0 &{}\text {otherwise.} \end{array}\right. } \end{aligned}$$

We also set for $i=1,2,3$

$$\begin{aligned} \begin{aligned}&F_i = \chi _i(x)F(u(x)), \quad {\widetilde{F}}_i = \chi _i(x)F(\zeta _Ru(x)), \\&f_i = \chi _i(x)f(u(x)), \quad {\widetilde{f}}_i = \chi _i(x)f(\zeta _Ru(x)), \\&{\widetilde{u}}_i=\chi _i(x)\zeta _R u(x). \end{aligned} \end{aligned}$$

Since $F_1={\widetilde{F}}_1$, $f_1={\widetilde{f}}_1$, $\widetilde{F}_3={\widetilde{f}}_3={\widetilde{u}}_3=0$, $L>1$, we have

$$\begin{aligned} \begin{aligned}&{1 \over 2}(\mathcal{D}'(u)-\mathcal{D}'(\zeta _Ru))\zeta _R u \\&\quad = \int _{{\mathbb {R}}^N}(I_\alpha *(F_1+F_2+F_3))(f_1{\widetilde{u}}_1+f_2\widetilde{u}_2) - \int _{{\mathbb {R}}^N}(I_\alpha *(F_1+{\widetilde{F}}_2))(f_1{\widetilde{u}}_1+{\widetilde{f}}_2{\widetilde{u}}_2) \\&\quad = \int _{{\mathbb {R}}^N}(I_\alpha *F_1)(f_2-{\widetilde{f}}_2){\widetilde{u}}_2 + \int _{{\mathbb {R}}^N}(I_\alpha *F_2)(f_1{\widetilde{u}}_1+f_2{\widetilde{u}}_2) \\&\qquad +\int _{{\mathbb {R}}^N}(I_\alpha *{\widetilde{F}}_2)(f_1{\widetilde{u}}_1+\widetilde{f}_2{\widetilde{u}}_2) + \int _{{\mathbb {R}}^N}(I_\alpha *F_3)(f_1\widetilde{u}_1+f_2{\widetilde{u}}_2). \end{aligned} \end{aligned}$$

Since $\Vert {F_2}\Vert _{2N\over N+\alpha }$, $\Vert {\widetilde{F}_2}\Vert _{2N\over N+\alpha }$, $\Vert {{\widetilde{f}}_2\widetilde{u}_2}\Vert _{2N\over N+\alpha }$, $\Vert {f_2{\widetilde{u}}_2}\Vert _{2N\over N+\alpha } \le C\sigma (\Vert {u}\Vert _{H^1(|{x}|\in [R,R+L])})^{N+\alpha \over 2N}$, $\Vert {F_1}\Vert _{2N\over N+\alpha }$, $\Vert {F_3}\Vert _{2N\over N+\alpha }$, $\Vert {f_1{\widetilde{u}}_1}\Vert _{2N\over N+\alpha }\le C\sigma (\Vert {u}\Vert _{H^1})^{N+\alpha \over 2N} \le C\sigma (M)^{N+\alpha \over 2N}$ and

$$\begin{aligned} \left| {\int _{{\mathbb {R}}^N}(I_\alpha *F_3)(f_1{\widetilde{u}}_1)}\right| \le D_L\Vert {F_3}\Vert _{L^{2N\over N+\alpha }(|{x}|\ge R+L)} \Vert {f_1{\widetilde{u}}_1}\Vert _{L^{2N\over N+\alpha }(|{x}|\le R)}, \end{aligned}$$

We can see that (i) holds. We can show (ii) in a similar way. $\square$

The above lemma gives a useful localization property of $\mathcal{D}(u)$.

Finally in this section we give the following lemma on the behavior of bounded Palais-Smale sequences, which will help us to get concentration-compactness type result in Sect. 4.

Lemma 2.4

There exists $\rho _1>0$ with the following property: if $(\varepsilon _j)_{j=1}^\infty \subset (0,1]$, a bounded sequence $(u_j)_{j=1}^\infty \subset H^1({\mathbb {R}}^N)$ and $(y_j)_{j=1}^\infty \subset {\mathbb {R}}^N$ satisfy

$$\begin{aligned} \begin{aligned}&I_{\varepsilon _j}'(u_j)\rightarrow 0 \quad \text {strongly in}\ (H^1({\mathbb {R}}^N))^*, \\&u_j(x+y_j) \rightharpoonup u_0 \quad \text {weakly in}\ H^1({\mathbb {R}}^N)\end{aligned} \end{aligned}$$

for some $u_0\in H^1({\mathbb {R}}^N)$ with $\Vert {u_0}\Vert _{H^1} \le \rho _1$, then $u_0=0$.

Proof

We set $v_j(x)=u_j(x+y_j)$. Let $L\in {\mathbb {N}}$. Since $(u_j)_{j=1}^\infty$ is bounded in $H^1({\mathbb {R}}^N)$, we have for $C>0$ independent of j and L

$$\begin{aligned} \sum _{i=1}^L \Vert {v_j}\Vert _{H^1(|{x}|\in [Li,L(i+1)])}^2 \le \Vert {v_j}\Vert _{H^1}^2 \le C. \end{aligned}$$

Thus there exists $i_j\in \{ 1,2,\cdots ,L\}$ such that $\Vert {v_j}\Vert _{H^1(|{x}|\in [Li_j,L(i_j+1)])}^2 \le {C\over L}$. Extracting a subsequence if necessary, we may assume that for any $L\in {\mathbb {N}}$ there exists $k_L\in \{ 1,2,\cdots ,L\}$ such that

$$\begin{aligned} \Vert {v_j}\Vert _{H^1(|{x}|\in [Lk_L,L(k_L+1)])}^2 \le {C\over L} \qquad \text {for all}\ j\in {\mathbb {N}}. \end{aligned}$$

(2.3)

Let $\zeta _R(s)$ be a function satisfying (2.1) and set

$$\begin{aligned} v_j^{(L)}(x) = \zeta _{Lk_L}(x) v_j(x). \end{aligned}$$

We have from (2.3)

$$\begin{aligned} \begin{aligned}&\left| {\int _{{\mathbb {R}}^N}\nabla v_j\nabla v_j^{(L)} + V(\varepsilon _jx+y_j)v_j v_j^{(L)} - \int _{{\mathbb {R}}^N}|{\nabla v_j^{(L)}}|^2 + V(\varepsilon _jx+y_j)(v_j^{(L)})^2}\right| \\&\quad \le \left| {\int _{{\mathbb {R}}^N}\nabla (v_j-v_j^{(L)})\nabla v_j^{(L)} + V(\varepsilon _jx+y_j)(v_j-v_j^{(L)}) v_j^{(L)}}\right| \le a_L, \\&|{(\mathcal{D}'(v_j)- \mathcal{D}'(v_j^{(L)}))v_j^{(L)}}| \le a_L, \end{aligned} \end{aligned}$$

where $a_L$ is independent of j and satisfies $a_L\rightarrow 0$ as $L\rightarrow \infty$. Here we apply Lemma 2.3 (i) with $R=Lk_L$ and L. Thus we have

$$\begin{aligned} I_{\varepsilon _j}'(v_j^{(L)}(x-y_j))(v_j^{(L)}(x-y_j))&=\int _{{\mathbb {R}}^N}|{\nabla v_j^{(L)}}|^2 + V(\varepsilon _jx+y_j)(v_j^{(L)})^2 - \mathcal{D}'(v_j^{(L)})v_j^{(L)}\nonumber \\&\le I_{\varepsilon _j}'(u_j)(v_j(x-y_j)) + 2a_L =o(1) + 2a_L. \end{aligned}$$

(2.4)

Since $v_j^{(L)}\rightarrow u_0^{(L)} \equiv \zeta _{Lk_L}(x) u_0(x)$ strongly in $L^p({\mathbb {R}}^N)$ for $p\in (2,{2N\over N-2})$ and $\Vert {I_{\varepsilon _j}'(u_j)}\Vert _{(H^1({\mathbb {R}}^N))^*}\rightarrow 0$,

$$\begin{aligned} \limsup _{j\rightarrow \infty } \int _{{\mathbb {R}}^N}|{\nabla v_j^{(L)}}|^2 + V(\varepsilon _jx+y_j)(v_j^{(L)})^2 \le \mathcal{D}'(u_0^{(L)})u_0^{(L)} + 2a_L. \end{aligned}$$

Let $\rho _0>0$ be the number given in Corollary 2.2. Since $\Vert {u_0^{(L)}}\Vert _{H^1} \le C\Vert {u_0}\Vert _{H^1}$, choosing $\rho _1>0$ small, we have for L large

$$\begin{aligned} \limsup _{j\rightarrow \infty } \Vert {v_j^{(L)}}\Vert _{H^1} \le \rho _0, \quad \text {provided}\ \Vert {u_0}\Vert _{H^1} \le \rho _1. \end{aligned}$$

By Corollary 2.2 and (2.4),

$$\begin{aligned} c_0\limsup _{j\rightarrow \infty } \Vert {v_j^{(L)}}\Vert _{H^1}^2 \le \limsup _{j\rightarrow \infty } I_{\varepsilon _j}'(v_j^{(L)}(x-y_j))(v_j^{(L)}(x-y_j)) \le 2a_L. \end{aligned}$$

Thus

$$\begin{aligned} \Vert {\zeta _{Lk_L}(x)u_0}\Vert _{H^1}^2 = \Vert {u_0^{(L)}}\Vert _{H^1}^2 \le \limsup _{j\rightarrow \infty } \Vert {v_j^{(L)}}\Vert _{H^1}^2 \le {2\over c_0} a_L. \end{aligned}$$

Since L is arbitrary, we have $u_0=0$. $\square$

3 Limit Problems

3.1 Limit Problems

For $a>0$ we define

$$\begin{aligned} L_a(u)={1 \over 2}\Vert {\nabla u}\Vert _2^2 +{a\over 2}\Vert {u}\Vert _2^2 -{1 \over 2}\mathcal{D}(u):\,H^1({\mathbb {R}}^N)\rightarrow {\mathbb {R}}. \end{aligned}$$

Critical points of $L_a(u)$ is a solution of

$$\begin{aligned} -\Delta u+au = (I_a*F(u))f(u) \quad \text {in}\ {\mathbb {R}}^N, \end{aligned}$$

(3.1)

which appears as a limit equation for (1.2). That is, for a family $(u_\varepsilon (x))$ of solutions of (1.2) and $(x_\varepsilon )\subset {\mathbb {R}}^N$ with $x_\varepsilon \rightarrow x_0$, if there exists a limit $v_0(x) = \lim _{\varepsilon \rightarrow 0} u_\varepsilon (x+{x_\varepsilon \over \varepsilon })$, then $v_0$ is a critical point of $L_{V(x_0)}(u)$, that is, a solution of (3.1) with $a=V(x_0)$. We denote by $E_a$ the least energy level for (3.1):

$$\begin{aligned} E_a = \inf \{ L_a(u):\, u\not =0,\, L_a'(u)=0\}. \end{aligned}$$

In [48], the existence of a least energy solution is proved under the conditions (f1)–(f3) and

• (f4’) there exists $s_0\in {\mathbb {R}}\setminus \{ 0\}$ such that $F(s_0)>0$.

They also proved that under (f1)–(f3), (f4’) every ground state solution of (3.1) is radially symmetric with respect to some point in ${\mathbb {R}}^N$. It is also shown that any solution of (3.1) satisfies the Pohozaev identity:

$$\begin{aligned} P_a(u)=0, \end{aligned}$$

where

$$\begin{aligned} P_a(u) = {N-2\over 2}\Vert {\nabla u}\Vert _2^2 +{N\over 2}a\Vert {u}\Vert _2^2 -{N+\alpha \over 2}\mathcal{D}(u). \end{aligned}$$

(3.2)

The least energy level $E_a$ is characterized as

$$\begin{aligned} E_a = \inf \{L_a(u):\, u\in H^1({\mathbb {R}}^N)\setminus \{ 0\},\, P_a(u)=0\}. \end{aligned}$$

(3.3)

For $c>0$ we set

$$\begin{aligned} \mathcal{S}_a^c =\{ u\in H^1({\mathbb {R}}^N)\setminus \{0\}:\, L_a'(u)=0,\, L_a(u)\le c,\, |{u(0)}|=\max _{x\in {\mathbb {R}}^N}|{u(x)}| \}. \end{aligned}$$

Arguing as in [48], we can show that

Lemma 3.1

$\mathcal{S}_a^c$ is compact in $H^1({\mathbb {R}}^N)$ provided $c< 2E_a$.

3.2 Scaling Argument for $L_a(u)$

As in [23], to see the scaling property of the limit function $L_a(u)$, we consider for $u\in H^1({\mathbb {R}}^N)\setminus \{ 0\}$

$$\begin{aligned} d(\lambda )=L_a(u(x/\lambda )) = {1 \over 2}\Vert {\nabla u}\Vert _2^2\lambda ^{N-2} +{a\over 2}\Vert {u}\Vert _2^2\lambda ^N -{1 \over 2}\mathcal{D}(u)\lambda ^{N+\alpha }:(0,\infty )\rightarrow {\mathbb {R}}. \end{aligned}$$

We have

• (i) $d(\lambda )\rightarrow +0$ as $\lambda \rightarrow +0$;

• (ii) $d(\lambda )\rightarrow -\infty$ as $\lambda \rightarrow \infty$;

• (iii) $d(\lambda )$ has a unique critical point $\lambda _0(u)$, which is a maximum of $d(\lambda )$;

• (iv) $d'(\lambda )=0$ if and only if $P_a(u(x/\lambda ))=0$.

In particular, we have

Proposition 3.2

For a least energy solution $\omega _0(x)$ of (3.1), that is, $L_a'(\omega _0)=0$, $L_a(\omega _0)=E_a$, we have

$$\begin{aligned} \begin{aligned}&L_a\left( \omega _0\left( {x\over s}\right) \right)< E_a \quad \text {for}\ s\in (0,\infty )\setminus \{ 1\}, \\&P_a\left( \omega _0\left( {x\over s}\right) \right) {\left\{ \begin{array}{ll} >0 &{}\text {for}\ s\in (0,1),\\ <0 &{}\text {for}\ s\in (1,\infty ). \end{array}\right. } \end{aligned} \end{aligned}$$

3.3 Center of Mass

Here we introduce a center of mass $\beta (u)$ in a neighborhood of a shifted compact set. We will use the following

Proposition 3.3

Let ${\widehat{D}}\subset H^1({\mathbb {R}}^N)\setminus \{ 0\}$ be a compact set. We set for $\rho >0$

$$\begin{aligned} \begin{aligned}&{\widetilde{D}} = \{ \omega (x-p):\, \omega \in {\widehat{D}},\, p\in {\mathbb {R}}^N\}, \\&{\widetilde{D}}_\rho =\{ u\in H^1({\mathbb {R}}^N):\, \mathop {\textrm{dist}}\nolimits _{H^1}(u, {\widetilde{D}})<\rho \}. \end{aligned} \end{aligned}$$

Then there exist $\rho _2>0$, $R_0>0$ and $C^1$-function $\beta :{\widetilde{D}}_{\rho _2}\rightarrow {\mathbb {R}}^N$ such that

• (i) For $u(x)=\omega (x-p)+\varphi (x)\in {\widetilde{D}}_{\rho _2}$ with $\omega \in {\widehat{D}}$, $p\in {\mathbb {R}}^N$, $\Vert {\varphi }\Vert _{H^1}<\rho _2$,

  $$\begin{aligned} |{\beta (u)-p}| \le R_0. \end{aligned}$$

• (ii) $\beta (u)$ is shift-invariant, that is,

  $$\begin{aligned} \beta (u(x-q)) = \beta (u) + q \end{aligned}$$

  for all $u\in {\widetilde{D}}_{\rho _2}$ and $q\in {\mathbb {R}}^N$.

• (iii) If u, $v\in {\widetilde{D}}_{\rho _2}$ satisfy

  $$\begin{aligned} u(x)=v(x) \quad \text {in}\ B(\beta (u),4R_0), \end{aligned}$$

  (3.4)

  then $\beta (u)=\beta (v)$.

• (iv) There exists $C>0$ independent of u such that

  $$\begin{aligned} \Vert {\beta '(u)}\Vert _{(H^1({\mathbb {R}}^N))^*} \le C \quad \text {for all}\ u\in {\widetilde{D}}_{\rho _2}. \end{aligned}$$

A similar center of mass is given in [8, 9], which is locally Lipschitz continuous. Here we modify and improve the argument in [8, 9] and give a center of mass $\beta (u)$, which is of class $C^1$.

Proof

We set $r_* = \inf _{\omega \in {\widehat{D}}} \Vert {\omega }\Vert _{H^1} > 0$. Since ${\widehat{D}}$ is compact, there exists $R_*>0$ such that

$$\begin{aligned} \Vert {\omega }\Vert _{H^1(|{x}|\le R_*)} \ge {2\over 3}r_*, \quad \Vert {\omega }\Vert _{H^1(|{x}|\ge R_*)} \le {1\over 6}r_* \quad \text {for all}\ \omega \in {\widehat{D}}. \end{aligned}$$

For $u=\omega (x-p)+\varphi (x)$ with

$$\begin{aligned} p\in {\mathbb {R}}^N, \ \omega \in {\widehat{D}} \ \text {and} \ \Vert {\varphi }\Vert _{H^1} < {1\over 6}r_* \end{aligned}$$

(3.5)

we have

$$\begin{aligned} \Vert {u(x)}\Vert _{H^1(|{x-p}|\le R_*)} \ge {1 \over 2}r_*, \qquad \Vert {u(x)}\Vert _{H^1(|{x-p}|\ge R_*)} \le {1\over 3} r_*. \end{aligned}$$

(3.6)

We set for $q\in {\mathbb {R}}^N$ and $u\in {\widetilde{D}}_{r_*/6}$

$$\begin{aligned} \Phi (q,u) = \int _{{\mathbb {R}}^N}\zeta _{R_*}(x-q)(|{\nabla u}|^2+ u^2)\, dx, \end{aligned}$$

where $\zeta _{R_*}(x-q)$ is introduced in (2.1). By (3.6), we have for $u(x)=\omega (x-p)+\varphi (x)\in \widetilde{D}_{r_*/6}$

$$\begin{aligned} \begin{aligned}&\Phi (p,u) \ge \left( {1 \over 2}r_*\right) ^2, \\&\Phi (q,u) \le \left( {1\over 3}r_*\right) ^2 \quad \text {for}\ |{q-p}|\ge 2R_*+1, \\&\Phi (q,u(x-q')) = \Phi (q-q',u(x)) \quad \text {for all}\ q,q'\in {\mathbb {R}}^N. \end{aligned} \end{aligned}$$

In fact, $\mathop {\textrm{supp}}\zeta _{R_*}(x-q)\subset \{x:\,|{x-p}|\ge R_*\}$ for $|{q-p}|\ge 2R_*+1$. We choose and fix a function $\psi (s)\in C^\infty ([0,\infty ),{\mathbb {R}})$ such that

$$\begin{aligned} \psi (s) ={\left\{ \begin{array}{ll} 1 &{}s\in [({1 \over 2}r_*)^2,\infty ), \\ 0 &{}s\in [0,({1\over 3}r_*)^2], \end{array}\right. } \qquad \psi (s)\in [0,1]\quad \text {for all}\ s\in {\mathbb {R}}. \end{aligned}$$

Then we have for $u=\omega (x-p)+\varphi (x)\in {\widetilde{D}}_{r_*/6}$ with (3.5)

$$\begin{aligned} \psi (\Phi (p,u)) =1 \quad \text {and}\quad \psi (\Phi (q,u)) =0 \quad \text {for}\ |{q-p}|\ge 2R_*+1. \end{aligned}$$

(3.7)

We set

$$\begin{aligned} \beta (u) = {\int _{{\mathbb {R}}^N}q\psi (\Phi (q,u))\, dq \over \int _{{\mathbb {R}}^N}\psi (\Phi (q,u))\, dq}:\, {\widetilde{D}}_{r_*/6}\rightarrow {\mathbb {R}}^N. \end{aligned}$$

Then we have

$$\begin{aligned}&|{\beta (u)-p}| \le 2R_*+1, \nonumber \\&\beta (u(x-q')) = \beta (u(x)) + q'. \end{aligned}$$

(3.8)

Thus, setting $R_0=2R_*+1$, $\rho _2=r_*/6$, we have (i)–(ii).

Next we prove (iii). We suppose that $u(x)=\omega (x-p)+\varphi (x)$, $v(x)=\omega '(x-p')+\varphi '(x)\in {\widetilde{D}}_{r_*/6}$ satisfy (3.4). By (3.7) and (3.8),

$$\begin{aligned} \mathop {\textrm{supp}}\psi (\Phi (\cdot ,u))\subset {\overline{B}}(p,R_0)\subset {\overline{B}}(\beta (u),2R_0). \end{aligned}$$

(3.9)

Similarly $\mathop {\textrm{supp}}\psi (\Phi (\cdot ,v))\subset {\overline{B}}(p',R_0)\subset {\overline{B}}(\beta (v),2R_0)$.

By (3.4), we have $v(x)=u(x)$ on ${\overline{B}}(p,R_0)$, from which we have $\psi (\Phi (p,v))=\psi (\Phi (p,u))=1$. Thus $p\in \mathop {\textrm{supp}}\psi (\Phi (\cdot ,v))$ and we have $|{p-p'}|\le R_0$. And thus $\mathop {\textrm{supp}}\psi (\Phi (\cdot ,v)) \subset {\overline{B}}(p',R_0)\subset {\overline{B}}(p, 2R_0)$. Since $v=u$ on ${\overline{B}}(p,3R_0)\subset {\overline{B}}(\beta (u),4R_0)$, we have $\psi (\Phi (\cdot ,v))=\psi (\Phi (\cdot ,u))$ on ${\mathbb {R}}^N$. Thus we have $\beta (v)=\beta (u)$.

Finally we prove (iv). We set $A=\int _{{\mathbb {R}}^N}\psi (\Phi (q,u))\, dq$. For $h\in H^1({\mathbb {R}}^N)$ we compute that

$$\begin{aligned} \begin{aligned} \beta '(u)h&={1\over A}\int _{{\mathbb {R}}^N}q\psi '(\Phi (q,u))\partial _u\Phi (q,u)h\,dq \\&\qquad -{1\over A^2}\int _{{\mathbb {R}}^N}q\psi (\Phi (q,u))\, dq \int _{{\mathbb {R}}^N}\psi '(\Phi (q,u))\partial _u\Phi (q,u)h\,dq \\&= {1\over A} \int _{{\mathbb {R}}^N}(q-\beta (u))\psi '(\Phi (q,u))\partial _u\Phi (q,u)h\,dq. \end{aligned} \end{aligned}$$

By (3.9),

$$\begin{aligned} \begin{aligned} |{\beta '(u)h}|&\le {2R_0\over A} \int _{{\mathbb {R}}^N}|{\psi '(\Phi (q,u))\partial _u\Phi (q,u)h}|\, dq \\&\le {2R_0\over A} |{B(\beta (u),2R_0)}|\, \Vert {\psi '}\Vert _\infty \max _{q\in {\overline{B}}(\beta (u),2R_0)}|{\partial _u\Phi (q,u)h}|. \end{aligned} \end{aligned}$$

Noting $|{\partial _u\Phi (q,u)h}|=2|{\int _{{\mathbb {R}}^N}\zeta _{R_*}(x-q)(\nabla u\nabla h+uh)}| \le 2\Vert {u}\Vert _{H^1}\Vert {h}\Vert _{H^1}$, we have (iv). $\square$

In the following sections, we develop a deformation argument for $I_\varepsilon (u)$ in ${\widetilde{D}}_{\rho _2}$ for a suitable choice of ${\widehat{D}}$.

4 A Neighborhood of Expected Solutions

In this section we set up a neighborhood of expected solutions, in which we will develop a deformation argument in Sect. 6.

4.1 A Neighborhood $\Omega$ of Concentrating Points

In this section, we show that we may assume the following (V4) in addition to (V1)–(V3) and (LM) (or (MP)).

• (V4) For any $p\in \Omega$, $2E_{V(p)} > E_{V_0}$.

In fact, since $E_a$ is a continuous function of $a\in (0,\infty )$, there exists $\alpha >0$ such that

$$\begin{aligned} 2E_{V_0-\alpha } > E_{V_0}. \end{aligned}$$

On the other hand, since V(x) is of class $C^N$, the set of critical values of V(x) is of measure 0 in ${\mathbb {R}}$ by Sard Theorem. Therefore we may assume $V_0-\alpha$ is a regular value of V(x). We set

$$\begin{aligned} \Omega _\alpha =\{ x\in \Omega :\, V(x)>V_0-\alpha \}. \end{aligned}$$

Then, V(x) satisfies (V1)–(V4).

We observe that if V(x) satisfies (LM) ((MP) respectively) in $\Omega$, then V(x) satisfies (LM) ((MP) respectively) in $\Omega _\alpha$. We show just for (MP).

We may assume $V(e_0)$, $V(e_1)<V_0-\alpha$. We set

$$\begin{aligned} \begin{aligned} M_i&= \{ x\in {\overline{\Omega }}:\, V(x)=V_0-\alpha , \ x \text { and } e_i \text { are path connected in} \\&\qquad \{ x\in \Omega :\, V(x) \le V_0-\alpha \} \} \quad \text {for}\ i=0,1, \\ {\widetilde{\Lambda }}&= \{c(\xi )\in ([0,1],\overline{\Omega _\alpha )}:\, c(0)\in M_0,\, c(1)\in M_1, \, V(c(\xi )) > V_0-\alpha \ \text {for}\ \xi \in (0,1)\}. \end{aligned} \end{aligned}$$

Then we can easily see that $V_0 =\inf _{c\in {\widetilde{\Lambda }}}\max _{\xi \in [0,1]} V(c(\xi ))$. Clearly there exists paths $(c_k)_{k=1}^\infty \subset \widetilde{\Lambda }$ with

$$\begin{aligned} c_k(0)\in M_0, \quad c_k(1)\in M_1, \quad \max _{\xi \in [0,1]} V(c_k(\xi )) \rightarrow V_0 \quad \text {as}\ k\rightarrow \infty . \end{aligned}$$

Since $M_0$, $M_1$ are compact, we may assume after extracting a subsequence

$$\begin{aligned} c_k(0) \rightarrow {\widetilde{e}}_0\in M_0, \quad c_k(1) \rightarrow {\widetilde{e}}_1\in M_1 \quad \text {as}\ k\rightarrow \infty .\ \end{aligned}$$

Choose $\widetilde{{\widetilde{e}}}_0$, $\widetilde{\widetilde{e}}_1\in \Omega _\alpha$ so that $\widetilde{{\widetilde{e}}}_0$ is close to ${\widetilde{e}}_0$ and $\widetilde{{\widetilde{e}}}_1$ is close to ${\widetilde{e}}_1$. Replacing $\Omega$, $e_0$, $e_1$, $\Lambda$ with $\Omega _\alpha$, $\widetilde{{\widetilde{e}}}_0$, $\widetilde{{\widetilde{e}}}_1$ and $\widetilde{{\widetilde{\Lambda }}} = \{ c(\xi )\in C([0,1],\Omega _\alpha ):\, c(0)=\widetilde{\widetilde{e}}_0,\, c(1)=\widetilde{{\widetilde{e}}}_1 \}$. we can see that (MP) holds.

4.2 A Neighborhood of Expected Solutions

In what follows, we assume (V1)–(V4) hold for $\Omega$ and $V_0$ is a critical value of V(x) in $\Omega$. We write

$$\begin{aligned} b=E_{V_0} \end{aligned}$$

and set

$$\begin{aligned} \mathcal{K}_b = \{(\xi ,\omega )\in \Omega \times H^1({\mathbb {R}}^N):\, \nabla V(\xi )=0,\, L_{V(\xi )}'(\omega )=0,\, L_{V(\xi )}(\omega )=b\}. \end{aligned}$$

We note that

$$\begin{aligned} \mathcal{K}_b = \{(\xi ,\omega )\in \Omega \times H^1({\mathbb {R}}^N):\, DL(\xi ,\omega )=0,\, L(\xi ,\omega )=b\}, \end{aligned}$$

where $D=(\partial _z,\partial _u)$ and

$$\begin{aligned} L(z,u)={1 \over 2}\Vert {\nabla u}\Vert _2^2 +{1 \over 2}V(z)\Vert {u}\Vert _2^2 -{1 \over 2}\mathcal{D}(u):\, {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}\rightarrow {\mathbb {R}}. \end{aligned}$$

We remark that L(z, u) appears as a limit functional for $I_\varepsilon (u)$. In fact, for $z\in {\mathbb {R}}^N$ and $u(x)\in H^1({\mathbb {R}}^N)$, we have

$$\begin{aligned} I_\varepsilon \left( u\left( x-{z\over \varepsilon }\right) \right) = {1 \over 2}\Vert {\nabla u}\Vert _2^2 +{1 \over 2}\int _{{\mathbb {R}}^N}V(\varepsilon x+z)u(x)^2 -{1 \over 2}\mathcal{D}(u) \rightarrow L(z,u) \quad \text {as}\ \varepsilon \rightarrow 0. \end{aligned}$$

In what follows, we denote the projections to the first and second components by

$$\begin{aligned} P_1(z,u)=z, \quad P_2(z,u)=u. \end{aligned}$$

Remark 4.1

• (i) We have

  $$\begin{aligned} \begin{aligned} \{ (\xi ,\omega )\in \,&\Omega \times H^1({\mathbb {R}}^N):\, V(\xi )=V_0,\, \nabla V(\xi )=0, \\&\omega \text { is a least energy solution of } L_{V_0}'(\omega )=0 \} \subset \mathcal{K}_b. \end{aligned} \end{aligned}$$

• (ii) Since $E_a> b=E_{V_0}$ for $a>V_0$, $(\xi ,\omega )\in \mathcal{K}_b$ implies $V(\xi )\le V_0$. Thus we have

  $$\begin{aligned} P_1\mathcal{K}_b \cap \partial \Omega =\emptyset , \qquad P_1\mathcal{K}_b \subset \{\xi \in \Omega :\, V(\xi )\le V_0\} \end{aligned}$$

  and $P_1\mathcal{K}_b$ is compact in $\Omega$ by the assumption (V3).

• (iii) If $(\xi ,\omega )\in \mathcal{K}_b$ satisfies $V(\xi )=V_0$, we have $L_{V(\xi )}(\omega )=b$, that is, $\omega$ is a least energy solution of $L_{V_0}(\cdot )$. On contrary, if $V(\xi )<V_0$, we have $L_{V(\xi )}(\omega )=b >E_{V(\xi )}$ and $\omega$ is not a least energy solution of $L_{V(\xi )}(\cdot )$.

We set $Q=[0,1]^N$ and

$$\begin{aligned} \widehat{\mathcal{K}}_b=\{ (\xi ,\omega )\in \mathcal{K}_b:\, \Vert {\omega }\Vert _{L^2(Q)} =\max _{n\in {\mathbb {Z}}^N}\Vert {\omega }\Vert _{L^2(n+Q)} \}. \end{aligned}$$

For $\varepsilon >0$ we set

$$\begin{aligned} {\widehat{K}}_b^{(\varepsilon )}= \left\{ \omega \left( x-{\xi \over \varepsilon }\right) :\, (\xi ,\omega )\in \widehat{\mathcal{K}}_b\right\} . \end{aligned}$$

and we try to find a critical point of $I_\varepsilon (u)$ in a neighborhood of ${\widehat{K}}_b^{(\varepsilon )}$. We introduce $\mathcal{K}_b$ and $\widehat{\mathcal{K}}_b$ to obtain necessary compactness properties, in particular, to show Proposition 4.5 below.

For our minimax argument, we also introduce

$$\begin{aligned}&{\widehat{S}}_b= \left\{ \omega \left( {x\over s}\right) :\, \omega \in P_2\widehat{\mathcal{K}}_b,\, s\in \left[ {1 \over 2},{3\over 2}\right] \right\} , \nonumber \\&\widehat{\mathcal{Z}}_b= \{ (\xi ,w):\, \xi \in \overline{\Omega },\, w\in {\widehat{S}}_b\},\nonumber \\&{\widehat{Z}}_b^{(\varepsilon )}= \left\{ \omega \left( x-{\xi \over \varepsilon }\right) :\, (\xi ,\omega )\in \widehat{\mathcal{Z}}_b\right\} . \end{aligned}$$

(4.1)

It holds

$$\begin{aligned} \widehat{\mathcal{K}}_b\subset \widehat{\mathcal{Z}}_b, \quad {\widehat{K}}_b^{(\varepsilon )}\subset {\widehat{Z}}_b^{(\varepsilon )}. \end{aligned}$$

By (V1)–(V4) and Lemma 3.1, we see

Lemma 4.2

$\widehat{\mathcal{K}}_b$ and $\widehat{\mathcal{Z}}_b$ are compact in ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$. ${\widehat{K}}_b^{(\varepsilon )}$ and ${\widehat{Z}}_b^{(\varepsilon )}$ are also compact in $H^1({\mathbb {R}}^N)$.

Here and in what follows we indicate compact sets by $\widehat{\cdot }$.

To describe neighborhoods, we introduce an $\varepsilon$-dependent distance $\mathop {\textrm{dist}}\nolimits _\varepsilon (\cdot ,\cdot )$ on $H^1({\mathbb {R}}^N)$ by

$$\begin{aligned} \mathop {\textrm{dist}}\nolimits _\varepsilon (u(x),v(x)) = \inf _{h\in {\mathbb {R}}^N}\left( |{h}|^2 + \left\| {u(x)-v\left( x-{h\over \varepsilon }\right) }\right\| _{H^1}^2\right) ^{1/2}. \end{aligned}$$

The $\varepsilon$-dependent distance $\mathop {\textrm{dist}}\nolimits _\varepsilon (\cdot ,\cdot )$ is a natural distance to consider concentration of a sequence $(u_{\varepsilon _j})_{j=1}^\infty \subset H^1({\mathbb {R}}^N)$, $\varepsilon _j\rightarrow 0$ to a limit profile $(\xi ,\omega )\in \widehat{\mathcal{K}}_b$ as $u_{\varepsilon _j}(x) \sim \omega (x-{\xi \over \varepsilon _j})$. In fact, introducing $H_\varepsilon :H^1({\mathbb {R}}^N)\rightarrow {\mathbb {R}}^N$ by

$$\begin{aligned} H_\varepsilon (u) = {1 \over 2}\int _{{\mathbb {R}}^N}\nabla V(\varepsilon x)u(x)^2, \end{aligned}$$

we have

Lemma 4.3

• (i) For $(\xi ,\omega )\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$, if $(u_j)_{j=1}^\infty \subset H^1({\mathbb {R}}^N)$, $\varepsilon _j\rightarrow 0$ satisfies

  $$\begin{aligned} \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}\left( u_j,\omega \left( x-{\xi \over \varepsilon _j}\right) \right) \rightarrow 0, \end{aligned}$$

  (4.2)

  then for $\varphi \in H^1({\mathbb {R}}^N)$

  $$\begin{aligned}&I_{\varepsilon _j}(u_j) \rightarrow L(\xi ,\omega ),\nonumber \\&I_{\varepsilon _j}'(u_j)\varphi \left( x-{\xi \over \varepsilon _j}\right) \rightarrow \partial _u L(\xi ,\omega )\varphi , \end{aligned}$$

  (4.3)
  $$\begin{aligned}&H_{\varepsilon _j}(u_j) \rightarrow {1 \over 2}\nabla V(\xi )\Vert {\omega }\Vert _2^2=\partial _z L(\xi ,\omega ). \end{aligned}$$

  (4.4)

• (ii) For $(\xi ,\omega )$, $(\xi ',\omega ')\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ with $\omega$, $\omega '\not =0$ and $\varepsilon _j\rightarrow 0$,

  $$\begin{aligned} \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}\left( \omega \left( x-{\xi \over \varepsilon _j}\right) ,\omega '\left( x-{\xi '\over \varepsilon _j}\right) \right) \rightarrow 0 \end{aligned}$$

  (4.5)

  holds if and only if

  $$\begin{aligned} \xi '=\xi \quad \text {and}\quad \omega '(x)=\omega (x-h_0) \ \text {for some}\ h_0\in {\mathbb {R}}^N. \end{aligned}$$

  (4.6)

Proof

(i) $\mathop {\textrm{dist}}\nolimits _{\varepsilon _j}(u_j(x),\omega (x-{\xi \over \varepsilon _j})) \rightarrow 0$ holds if and only if there exists $(h_j)_{j=1}^\infty \subset {\mathbb {R}}^N$ and $({\varphi _j})_{j=1}^\infty \subset H^1({\mathbb {R}}^N)$ such that

$$\begin{aligned} h_j\rightarrow 0, \quad \Vert {\varphi _j}\Vert _{H^1}\rightarrow 0 \end{aligned}$$

and

$$\begin{aligned} u_j(x)=\omega \left( x-{\xi +h_j\over \varepsilon _j}\right) + \varphi _j\left( x-{\xi +h_j\over \varepsilon _j}\right) . \end{aligned}$$

Thus

$$\begin{aligned} \begin{aligned} I_{\varepsilon _j}(u_j)&= {1 \over 2}\Vert {\nabla \omega +\nabla \varphi _j}\Vert _{H^1}^2 +{1 \over 2}\int _{{\mathbb {R}}^N}V(\varepsilon _jx + \xi +h_j)|{\omega (x)+\varphi _j(x)}|^2\\&\quad -{1 \over 2}\mathcal{D}(\omega +\varphi _j) \\&\rightarrow L(\xi ,\omega ). \end{aligned} \end{aligned}$$

(4.3) and (4.4) hold in a similar way. (ii) can be shown easily. $\square$

We also note that $\mathop {\textrm{dist}}\nolimits _\varepsilon (\cdot ,\cdot )$ is weaker than $H^1$-distance, namely there exist sequences $(u_j)_{j=1}^\infty$, $(v_j)_{j=1}^\infty \subset H^1({\mathbb {R}}^N)$ such that for $\varepsilon _j\rightarrow 0$

$$\begin{aligned} \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}(u_j,v_j)\rightarrow 0, \qquad \liminf _{j\rightarrow \infty }\Vert {u_j-v_j}\Vert _{H^1} > 0. \end{aligned}$$

(4.7)

In fact, for $\omega \not =0$, setting $u_j(x)=\omega (x-{p_1\over \sqrt{\varepsilon }_j})$, $v_j(x)=\omega (x)$, where $p_1=(1,0,\cdots ,0)$, we have (4.7).

Lemma 4.3 (i) shows that for $(\xi ,\omega )\in \widehat{\mathcal{K}}_b$, $(u_j)_{j=1}^\infty$ satisfying (4.2) is an $\varepsilon$-dependent Palais-Smale type sequence with the limit profile $(\xi ,\omega )$. Conversely, in Proposition 4.5 below, we study the convergence of $\varepsilon$-dependent Palais-Smale type sequences with respect to the distance $\mathop {\textrm{dist}}\nolimits _\varepsilon (\cdot ,\cdot )$.

We set for $\rho >0$

$$\begin{aligned} \begin{aligned} N_\rho ^{(\varepsilon )}&=\{ u\in H^1({\mathbb {R}}^N):\, \mathop {\textrm{dist}}\nolimits _\varepsilon (u,{\widehat{K}}_b^{(\varepsilon )})<\rho \} \\&= \left\{ \omega \left( x-{\xi +h\over \varepsilon }\right) +\varphi \left( x-{\xi +h\over \varepsilon }\right) :\, (\xi ,\omega )\in \widehat{\mathcal{K}}_b,\, |{h}|^2+\Vert {\varphi }\Vert _{H^1}^2< \rho ^2\right\} , \\ A_\rho ^{(\varepsilon )}&=\{ u\in H^1({\mathbb {R}}^N):\, \mathop {\textrm{dist}}\nolimits _\varepsilon (u,{\widehat{Z}}_b^{(\varepsilon )})<\rho \} \\&= \left\{ \omega \left( x-{\xi +h\over \varepsilon }\right) +\varphi \left( x-{\xi +h\over \varepsilon }\right) :\, \omega \in {\widehat{S}}_b,\, \xi \in {\overline{\Omega }},\, |{h}|^2+\Vert {\varphi }\Vert _{H^1}^2 < \rho ^2\right\} . \end{aligned} \end{aligned}$$

These sets are uniformly bounded with respect to $\varepsilon \in (0,1]$ and we have

$$\begin{aligned} N_\rho ^{(\varepsilon )} \subset A_\rho ^{(\varepsilon )}. \end{aligned}$$

In what follows, for suitable $0<\rho <\rho '$ we develop a deformation argument in $A_{\rho '}^{(\varepsilon )}$ to find a critical point in $N_\rho ^{(\varepsilon )}$.

Remark 4.4

The reason we introduce $A_\rho ^{(\varepsilon )}$ is to construct neighborhoods which are suitable for our deformation arguments. Our neighborhood $A_\rho ^{(\varepsilon )}$ includes a suitable initial path in $H^1({\mathbb {R}}^N)$ which is related to a minimax argument in $\Omega \subset {\mathbb {R}}^N$. See Sect. 7.1 below. Our another neighborhood $N_\rho ^{(\varepsilon )}$ is precisely an $\varepsilon$-neighborhood of expected solutions with the profile in $\widehat{\mathcal{K}}_b$.

4.3 Concentration-Compactness Type Results

In this section we give an $\varepsilon$-dependent concentration-compactness type results, which will be useful to develop deformation theory in Sect. 6.

Proposition 4.5

There exists $\rho _3>0$ such that if $(\varepsilon _j)_{j=1}^\infty \subset (0,1]$ and $(u_j)_{j=1}^\infty \subset H^1({\mathbb {R}}^N)$ satisfy $\varepsilon _j\rightarrow 0$, $u_j\in A_{\rho _3}^{(\varepsilon _j)}$ and

$$\begin{aligned}&I_{\varepsilon _j}(u_j) \rightarrow b, \end{aligned}$$

(4.8)

$$\begin{aligned}&I_{\varepsilon _j}'(u_j) \rightarrow 0 \quad \text {strongly in}\ (H^1({\mathbb {R}}^N))^*, \end{aligned}$$

(4.9)

$$\begin{aligned}&H_{\varepsilon _j}(u_j) \rightarrow 0 \quad \text {in}\ {\mathbb {R}}^N \end{aligned}$$

(4.10)

as $j\rightarrow \infty$, then

$$\begin{aligned} \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}(u_j,{\widehat{K}}_b^{(\varepsilon _j)}) \rightarrow 0 \quad \text {as}\ j\rightarrow \infty . \end{aligned}$$

In particular, for any $\rho >0$ there exists $j_\rho \in {\mathbb {N}}$ such that

$$\begin{aligned} u_j\in \overline{N_\rho ^{(\varepsilon _j)}} \quad \text {for}\ j\ge j_\rho . \end{aligned}$$

Remark 4.6

To show the existence of a family concentrating at a local minimum of V(x), in [23] we obtained a similar result for $(u_j)_{j=1}^\infty \subset N_{\rho _3}^{(\varepsilon _j)}$ but without the assumption (4.10). To study concentration at local maxima and saddle points, we need (4.10). In fact, if $(\xi ,\omega )\in \Omega \times H^1({\mathbb {R}}^N)$ satisfies

$$\begin{aligned} L(\xi ,\omega )=b, \quad \partial _u L(\xi ,\omega )=0,\quad \Vert {\omega }\Vert _{L^2(Q)}=\max _{n\in {\mathbb {Z}}^N} \Vert {\omega }\Vert _{L^2(n+Q)}, \end{aligned}$$

then $u_j(x)=\omega (x-{\xi \over \varepsilon _j})$ with $\varepsilon _j\rightarrow 0$ satisfies (4.8) and (4.9). However we don’t have $\nabla V(\xi )=0$ and the limit set

$$\begin{aligned} \{ (\xi ,\omega ):\, L(\xi ,\omega )=b,\, \partial _u L(\xi ,\omega )=0,\, \Vert {\omega }\Vert _{L^2(Q)}=\max _{n\in {\mathbb {Z}}^N} \Vert {\omega }\Vert _{L^2(n+Q)}\}\quad \end{aligned}$$

(4.11)

is not compact in $\Omega \times H^1({\mathbb {R}}^N)$ in general.

We note that if b is corresponding to local minimum, $L(\xi ,\omega )=b$, $\partial _u L(\xi ,\omega )=0$ imply $V(\xi )=V_0\,(= \inf _{x\in \Omega } V(x))$, $L(\xi ,\omega )=E_{V_0}$ and the set defined in (4.11) is compact.

Proof of Proposition 4.5

For $\rho '>0$ suppose that $(\varepsilon _j)_{j=1}^\infty$, $(u_j)_{j=1}^\infty$ satisfy $\varepsilon _j\rightarrow 0$, $u_j\in A_{\rho '}^{(\varepsilon _j)}$ and (4.8)–(4.10). Since $u_j\in A_{\rho '}^{(\varepsilon _j)}$, there exist $(\xi _j,\omega _j)\in \widehat{\mathcal{Z}}_b$, $\varphi _j\in H^1({\mathbb {R}}^N)$ and $h_j\in {\mathbb {R}}^N$ such that

$$\begin{aligned}&u_j(x) = \omega _j\left( x-{\xi _j+h_j\over \varepsilon _j}\right) +\varphi _j\left( x-{\xi _j+h_j\over \varepsilon _j}\right) , \end{aligned}$$

(4.12)

$$\begin{aligned}&\Vert {\varphi _j}\Vert _{H^1}< \rho ', \qquad |{h_j}| < \rho '. \end{aligned}$$

(4.13)

Extracting a subsequence if necessary, we may assume for some $(\xi _0,\omega _0)\in \widehat{\mathcal{Z}}_b$, $\varphi _0\in H^1({\mathbb {R}}^N)$ and $h_0\in {\mathbb {R}}^N$ such that

$$\begin{aligned} \begin{aligned}&\xi _j\rightarrow \xi _0, \qquad h_j\rightarrow h_0, \\&\omega _j\rightarrow \omega _0 \quad \text {strongly in}\ H^1({\mathbb {R}}^N), \\&\varphi _j\rightharpoonup \varphi _0 \quad \text {weakly in}\ H^1({\mathbb {R}}^N). \end{aligned} \end{aligned}$$

We set

$$\begin{aligned}&{\widetilde{\xi }}_j\equiv \xi _j + h_j \rightarrow {\widetilde{\xi }}_0\equiv \xi _0+h_0,\nonumber \\&{\widetilde{\omega }}_j(x) \equiv \omega _j(x) +\varphi _j(x) \rightharpoonup {\widetilde{\omega }}_0(x) = \omega _0+\varphi _0 \quad \text {weakly in}\ H^1({\mathbb {R}}^N). \end{aligned}$$

(4.14)

Suppose $\rho '\in (0,\rho _1)$, where $\rho _1>0$ is given by Lemma 2.4. Then we have

Step 1: ${\widetilde{\omega }}_j(x)\rightarrow {\widetilde{\omega }}_0(x)$ strongly in $H^1({\mathbb {R}}^N)$.

It suffices to show that

$$\begin{aligned} \sup _{n\in {\mathbb {Z}}^N} \Vert {{\widetilde{\omega }}_j-{\widetilde{\omega }}_0}\Vert _{L^2(n+Q)} \rightarrow 0 \quad \text {as}\ j\rightarrow \infty . \end{aligned}$$

(4.15)

Since $({\widetilde{\omega }}_j)_{j=1}^\infty$ is bounded in $H^1({\mathbb {R}}^N)$, (4.15) implies for $p\in (2,{2N\over N-2})$

$$\begin{aligned} \begin{aligned}&{\widetilde{\omega }}_j\rightarrow {\widetilde{\omega }}_0 \quad \text {strongly in}\ L^p({\mathbb {R}}^N), \\&\mathcal{D}'({\widetilde{\omega }}_j){\widetilde{\omega }}_j \rightarrow \mathcal{D}'({\widetilde{\omega }}_0){\widetilde{\omega }}_0 \quad \text {as}\ j\rightarrow \infty . \end{aligned} \end{aligned}$$

It follows from

$$\begin{aligned} I_{\varepsilon _j}'\left( {\widetilde{\omega }}_j\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) \right) {\widetilde{\omega }}_j\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) \rightarrow 0, \qquad I_{\varepsilon _j}'\left( {\widetilde{\omega }}_j\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) \right) {\widetilde{\omega }}_0\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) \rightarrow 0 \end{aligned}$$

that

$$\begin{aligned} \begin{aligned}&\Vert {\nabla {\widetilde{\omega }}_j}\Vert _2^2 + \int _{{\mathbb {R}}^N}V(\varepsilon _jx+{\widetilde{\xi }}_j) {\widetilde{\omega }}_j^2 = \mathcal{D}'({\widetilde{\omega }}_j){\widetilde{\omega }}_j + o(1) = \mathcal{D}'({\widetilde{\omega }}_0){\widetilde{\omega }}_0 + o(1), \\&\Vert {\nabla {\widetilde{\omega }}_0}\Vert _2^2 + \int _{{\mathbb {R}}^N}V(\varepsilon _jx+{\widetilde{\xi }}_j) {\widetilde{\omega }}_0^2 = \mathcal{D}'({\widetilde{\omega }}_0){\widetilde{\omega }}_0 + o(1). \end{aligned} \end{aligned}$$

And thus ${\widetilde{\omega }}_j\rightarrow {\widetilde{\omega }}_0$ strongly in $H^1({\mathbb {R}}^N)$.

If (4.15) does not hold, there exists $(n_j)_{j=1}^\infty \subset {\mathbb {Z}}^N$ such that

$$\begin{aligned} \Vert {{\widetilde{\omega }}_j-{\widetilde{\omega }}_0}\Vert _{L^2(n_j+Q)} \not \rightarrow 0. \end{aligned}$$

(4.16)

By (4.14), we have $|{n_j}|\rightarrow \infty$. Thus letting ${\widetilde{\omega }}_j(x+n_j) \rightharpoonup \widetilde{{\widetilde{\omega }}}_0(x)$ weakly in $H^1({\mathbb {R}}^N)$, we have from (4.13), (4.16) that $\widetilde{{\widetilde{\omega }}}_0\not =0$ and

$$\begin{aligned} \Vert {\widetilde{{\widetilde{\omega }}}_0}\Vert _{H^1}\le \rho '. \end{aligned}$$

(4.17)

On the other hand, since ${\widetilde{\omega }}_j(x+n_j)=u_j(x+{{\widetilde{\xi }}_j\over \varepsilon _j}+n_j)$ and $I_{\varepsilon _j}'(u_j)\rightarrow 0$ strongly in $(H^1({\mathbb {R}}^N))^*$, Lemma 2.4 and (4.17) imply $\widetilde{{\widetilde{\omega }}}_0=0$, which is in contradiction.

Step 2: $\nabla V({\widetilde{\xi }}_0)=0$.

We have

$$\begin{aligned} \begin{aligned} H_{\varepsilon _j}(u_j)&= {1 \over 2}\int _{{\mathbb {R}}^N}\nabla V(\varepsilon _j x)u_j(x)^2 = {1 \over 2}\int _{{\mathbb {R}}^N}\nabla V(\varepsilon _j x){\widetilde{\omega }}_j\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) ^2 \\&= {1 \over 2}\int _{{\mathbb {R}}^N}\nabla V(\varepsilon _j x+{\widetilde{\xi }}_j ){\widetilde{\omega }}_j(x)^2 \rightarrow {1 \over 2}\nabla V({\widetilde{\xi }}_0)\Vert {{\widetilde{\omega }}_0}\Vert _2^2 \quad \text {as}\ j\rightarrow \infty \end{aligned} \end{aligned}$$

and thus (4.10) implies $\nabla V({\widetilde{\xi }}_0)=0$.

Step 3: $DL({\widetilde{\xi }}_0,{\widetilde{\omega }}_0)=0$ and $L({\widetilde{\xi }}_0,{\widetilde{\omega }}_0)=b$.

For any $\varphi \in C_0^\infty ({\mathbb {R}}^N)$, we have

$$\begin{aligned} \begin{aligned} I_{\varepsilon _j}'(u_j)\varphi \left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right)&= \int _{{\mathbb {R}}^N}\nabla {\widetilde{\omega }}_j\nabla \varphi + V(\varepsilon _jx+{\widetilde{\xi }}_j){\widetilde{\omega }}_j\varphi -\mathcal{D}'({\widetilde{\omega }}_j)\varphi \\&\rightarrow \int _{{\mathbb {R}}^N}\nabla {\widetilde{\omega }}_0\nabla \varphi + V({\widetilde{\xi }}_0){\widetilde{\omega }}_0\varphi -\mathcal{D}'({\widetilde{\omega }}_0)\varphi . \end{aligned} \end{aligned}$$

Thus (4.9) implies $\partial _u L({\widetilde{\xi }}_0,{\widetilde{\omega }}_0)=0$. It is easily seen that (4.8) implies $L({\widetilde{\xi }}_0,{\widetilde{\omega }}_0)=b$.

Step 4: For $\rho '>0$ small, $\mathop {\textrm{dist}}\nolimits _{\varepsilon _j}(u_j,{\widehat{K}}_b^{(\varepsilon _j)})\rightarrow 0$

It is clear that ${\widetilde{\xi }}_j=\xi _j+h_j$ is in a $\rho '$-neighborhood of $\Omega$ and thus so is ${\widetilde{\xi }}_0$. Since $\nabla V(x)\not =0$ on $\partial \Omega$, we have $({\widetilde{\xi }}_0,{\widetilde{\omega }}_0)\in \mathcal{K}_b$ if $\rho '>0$ is sufficiently small. Thus there exists $h_0\in {\mathbb {R}}^N$ such that ${\widehat{\omega }}_0(x)={\widetilde{\omega }}_0(x-h_0)$ satisfies $({\widetilde{\xi }}_0,{\widehat{\omega }}_0)\in \widehat{\mathcal{K}}_b$. We have

$$\begin{aligned} \begin{aligned} \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}(u_j,{\widehat{K}}_b^{(\varepsilon _j)})&\le \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}\left( u_j(x),{\widehat{\omega }}_0\left( x-{{\widetilde{\xi }}_0\over \varepsilon _j}\right) \right) \\&= \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}\left( {\widetilde{\omega }}_j\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) ,{\widehat{\omega }}_0\left( x-{{\widetilde{\xi }}_0\over \varepsilon _j}\right) \right) \\&\le \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}\left( {\widetilde{\omega }}_j\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) ,{\widetilde{\omega }}_0\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) \right) \\&\quad + \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}\left( {\widetilde{\omega }}_0\left( x-{{\widetilde{\xi }}_j\over \varepsilon _j}\right) ,{\widehat{\omega }}_0\left( x-{{\widetilde{\xi }}_0\over \varepsilon _j}\right) \right) \\&\le \Vert {{\widetilde{\omega }}_j-{\widetilde{\omega }}_0}\Vert _{H^1} + \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}\left( {\widehat{\omega }}_0\left( x-{{\widetilde{\xi }}_j-\varepsilon _j h_0\over \varepsilon _j}\right) ,{\widehat{\omega }}_0\left( x-{{\widetilde{\xi }}_0\over \varepsilon _j}\right) \right) \\&\le \Vert {{\widetilde{\omega }}_j-{\widetilde{\omega }}_0}\Vert _{H^1} + |{{\widetilde{\xi }}_j-\varepsilon _j h_0-{\widetilde{\xi }}_0}| \rightarrow 0 \quad \text {as}\ j\rightarrow \infty . \end{aligned} \end{aligned}$$

Thus choosing $\rho _3>0$ small, the proof is completed. $\square$

Next we show that $I_\varepsilon (u)$ satisfies the Palais-Smale type condition in $A_{\rho _1}^{(\varepsilon )}$ for $\varepsilon \in (0,1]$ fixed.

Proposition 4.7

Let $\rho _1>0$ be the number given in Lemma 2.4. For $\varepsilon \in (0,1]$ fixed, $I_\varepsilon (u)$ satisfies the Palais-Smale type condition in $A_{\rho _1}^{(\varepsilon )}$. That is, if $(u_j)_{j=1}^\infty \subset A_{\rho _1}^{(\varepsilon )}$ satisfies

$$\begin{aligned} (H_\varepsilon (u_j),I_\varepsilon '(u_j)) \rightarrow 0 \quad \text {strongly in} \ ({{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)})^*, \end{aligned}$$

(4.18)

then $(u_j)_{j=1}^\infty$ has a strongly convergent subsequence in $H^1({\mathbb {R}}^N)$. Moreover, after extracting a subsequence if necessary, assume $u_j\rightarrow u_0$ strongly as $j\rightarrow \infty$. Then $u_0$ satisfies $I_\varepsilon '(u_0)=0$ and

$$\begin{aligned} H_\varepsilon (u_0) = 0. \end{aligned}$$

(4.19)

Proof

Since $(u_j)_{j=1}^\infty \subset A_{\rho _1}^{(\varepsilon )}$, there exist $(\xi _j,\omega _j)\in \widehat{\mathcal{Z}}_b$, $h_j\in {\mathbb {R}}^N$ and $\varphi _j\in H^1({\mathbb {R}}^N)$ such that

$$\begin{aligned} \begin{aligned}&u_j(x)=\omega _j\left( x-{\xi _j+h_j\over \varepsilon _j}\right) + \varphi _j\left( x-{\xi _j+h_j\over \varepsilon _j}\right) , \\&|{h_j}|\le \rho _1, \quad \Vert {\varphi _j}\Vert _{H^1} \le \rho _1. \end{aligned} \end{aligned}$$

Extracting a subsequence if necessary, we may assume for some $(\xi _0,\omega _0)\in \widehat{\mathcal{K}}_b$, $\varphi _0\in H^1({\mathbb {R}}^N)$ and $h_0\in {\mathbb {R}}^N$

$$\begin{aligned} \begin{aligned}&\xi _j\rightarrow \xi _0, \qquad h_j\rightarrow h_0, \\&\omega _j\rightarrow \omega _0 \quad \text {strongly in}\ H^1({\mathbb {R}}^N),\\&\varphi _j\rightharpoonup \varphi _0 \quad \text {weakly in}\ H^1({\mathbb {R}}^N). \end{aligned} \end{aligned}$$

Using Lemma 2.4 and arguing as in Step 1 of the proof of Proposition 4.5, we have the strong convergence of $(u_j)$. (4.19) follows from $H_\varepsilon (u_j)\rightarrow 0$. $\square$

Remark 4.8

In Proposition 4.7, the condition (4.18) can be relaxed to

$$\begin{aligned} I_\varepsilon '(u_j) \rightarrow 0 \quad \end{aligned}$$

To see this fact, first we remark that $I_\varepsilon '(u_0)=0$ implies $H_\varepsilon (u_0)=0$. Indeed, from the regularity argument (c.f. [47, 50]), it follows from $I_\varepsilon '(u_0)=0$ that $u_0\in H^2({\mathbb {R}}^N)$. On the other hand, we have for $j\in \{1,2,\cdots ,N\}$

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^N}\nabla u_0\nabla u_{0x_j} = {1 \over 2}\int _{{\mathbb {R}}^N}\partial _{x_j}(|{\nabla u_0}|^2) =0, \\&\int _{{\mathbb {R}}^N}V(\varepsilon x)u_0 u_{0x_j} = {1 \over 2}\int _{{\mathbb {R}}^N}V(\varepsilon x) \partial _{x_j}(u_0^2) = -{\varepsilon \over 2}\int _{{\mathbb {R}}^N}{\partial V\over \partial x_j}(\varepsilon x)u_0^2, \\&\mathcal{D}'(u)(u_{0x_j}) = \int _{{\mathbb {R}}^N}(I_\alpha *F(u_0))F'(u_0)u_{0x_j} = \int _{{\mathbb {R}}^N}(I_\alpha *F(u_0))(F(u_0))_{x_j} = 0. \end{aligned} \end{aligned}$$

Thus $I_\varepsilon '(u_0)=0$ implies $\int _{{\mathbb {R}}^N}{\partial V\over \partial x_j}(\varepsilon x)u_0^2 = 0$ for $j=1,2,\cdots ,N$. That is,

$$\begin{aligned} H_\varepsilon (u_0)= {1 \over 2}\int _{{\mathbb {R}}^N}\nabla V(\varepsilon x)u_0^2 =0. \end{aligned}$$

If $I_\varepsilon '(u_j)\rightarrow 0$ strongly in $(H^1({\mathbb {R}}^N))^*$, from the proof of Proposition 4.7 there exists a strongly convergent subsequence $(u_{j_k})_{k=1}^\infty$. Let $u_{j_k}\rightarrow u_0$ in $H^1({\mathbb {R}}^N)$. Then we have $I_\varepsilon '(u_0)=0$, $H_\varepsilon (u_{j_k})\rightarrow H_\varepsilon (u_0)$. Since $I_\varepsilon '(u_0)=0$ implies $H_\varepsilon (u_0)=0$, we have $H_\varepsilon (u_{j_k})\rightarrow 0$. Thus we have (4.18).

4.4 A Choice of Neighborhoods and Gradient Estimates

We choose ${\rho _{*\!*}}>0$ small so that in a neighborhood $A_{{\rho _{*\!*}}}^{(\varepsilon )}$ of ${\widehat{K}}_b^{(\varepsilon )}$, we can develop a deformation argument for a proof of our main result.

We set

$$\begin{aligned} \begin{aligned}&{\widetilde{S}}_b =\{ w(x-p):\, w\in {\widehat{S}}_b, \, p\in {\mathbb {R}}^N\}, \\&{\widetilde{S}}_{b,\rho }=\{ u\in H^1({\mathbb {R}}^N):\, \mathop {\textrm{dist}}\nolimits _{H^1}(u,\widetilde{S}_b)<\rho \} \quad \text {for}\ \rho >0. \end{aligned} \end{aligned}$$

Here ${\widehat{S}}_b$ is defined in (4.1). Applying the argument in Sect. 3.3 with ${\widehat{D}}={\widehat{S}}_b$, $\widetilde{D}={\widetilde{S}}_b$ and ${\widetilde{D}}_\rho ={\widetilde{S}}_{b,\rho }$, we can define the center of mass:

$$\begin{aligned} \beta :\, {\widetilde{S}}_{b,\rho _2}\rightarrow {\mathbb {R}}^N \quad \text {for small}\ \rho _2>0. \end{aligned}$$

We choose and fix ${\rho _{*}}$, ${\rho _{*\!*}}>0$ such that

$$\begin{aligned} 0< 16{\rho _{*}}< {\rho _{*\!*}}< \min \left\{ {1\over 6}\rho _0, \rho _1, \rho _2,\rho _3 \right\} , \end{aligned}$$

(4.20)

where $\rho _2$ is given above and $\rho _0$ ($\rho _1$, $\rho _3$ respectively) is given in Corollary 2.2 (Lemma 2.4, Proposition 4.5 respectively). We will use relation $16{\rho _{*}}<{\rho _{*\!*}}$ later in the proof of Lemma 6.9. We note that the center of mass $\beta (u)$ is defined on $A_{{\rho _{*\!*}}}^{(\varepsilon )}$ and

$$\begin{aligned} \mathop {\textrm{dist}}\nolimits _{{\mathbb {R}}^N}(\varepsilon \beta (u),\Omega ) < \varepsilon R_0+{\rho _{*\!*}}\quad \text {for}\ u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}. \end{aligned}$$

(4.21)

In fact, by the definition of $A_{{\rho _{*\!*}}}^{(\varepsilon )}$, we have for some $\xi \in {\overline{\Omega }}$, $\omega \in {\widehat{S}}_b$, $h\in {\mathbb {R}}^N$

$$\begin{aligned} |{h}|^2 + \left\| {u(x)-\omega \left( x-{\xi +h\over \varepsilon }\right) }\right\| _{H^1}^2 < {\rho _{*\!*}}^2. \end{aligned}$$

Thus by Proposition 3.3 (i) we have (4.21).

By Propositions 4.5 and 4.7, we have the following estimates.

Proposition 4.9

For $0<{\rho _{*}}<{\rho _{*\!*}}$ with (4.20). Then we have

• (i) There exist $\varepsilon _0>0$, $\nu _0>0$ and $\delta _0>0$ with the following properties: For $\varepsilon \in (0,\varepsilon _0]$

  $$\begin{aligned} \Vert {(H_\varepsilon (u),I_\varepsilon '(u))}\Vert _{({{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)})^*} \equiv \left( |{H_\varepsilon (u)}|^2 + \Vert {I_\varepsilon '(u)}\Vert _{(H^1({\mathbb {R}}^N))^*}^2\right) ^{1/2} \ge \nu _0 \end{aligned}$$

  for all $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}{\setminus } \overline{N_{{\rho _{*}}}^{(\varepsilon )}}$ with $I_\varepsilon (u)\in [b-\delta _0, b+\delta _0]$.

• (ii) Suppose that for some $\varepsilon \in (0,\varepsilon _0]$

  $$\begin{aligned} (H_\varepsilon (u),I_\varepsilon '(u)) \not = 0 \quad \text {for all}\ u\in \overline{N_{{\rho _{*}}}^{(\varepsilon )}}\ \text {with} \ I_\varepsilon (u)\in [b-\delta _0,b+\delta _0].\quad \end{aligned}$$

  (4.22)

  Then there exists $\nu _\varepsilon >0$ such that

  $$\begin{aligned} \Vert {(H_\varepsilon (u),I_\varepsilon '(u))}\Vert _{({{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)})^*} \ge \nu _\varepsilon , \end{aligned}$$

  (4.23)

  for $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$ with $I_\varepsilon (u) \in [b-\delta _0, b+\delta _0]$.

In what follows we assume without loss of generality $\nu _\varepsilon \le \nu _0$.

Proof

(i), (ii) follow from Propositions 4.5 and 4.7 easily. $\square$

We fix $\varepsilon _0$, $\nu _0>0$ and $\delta _0>0$ given in Proposition 4.9.

Remark 4.10

(4.22) can be replaced with $I_\varepsilon '(u)\not =0$. We note that $I_\varepsilon '(u)=0$ implies $H_\varepsilon (u)=0$ (see Remark 4.8). (4.23) can be replaced by

$$\begin{aligned} \Vert {I_\varepsilon '(u)}\Vert _{(H^1({\mathbb {R}}^N))^*} \ge \nu _\varepsilon . \end{aligned}$$

In the following Sect. 5, we develop a special deformation argument for $I_\varepsilon (u)$.

5 A Functional Corresponding to the Tail of a Function

5.1 Functional $T_\varepsilon (u)$

To find a critical point of $I_\varepsilon (u)$ in a neighborhood $N_\rho ^{(\varepsilon )}$ of expected solutions, it is important to control the size of u outside of a ball $B(\beta (u),{4\over \sqrt{\varepsilon }})$.

We set for $u\in {\widetilde{S}}_{b,\rho _2}$ and $\varepsilon >0$

$$\begin{aligned} T_\varepsilon (u)=\int _{{\mathbb {R}}^N}{\widetilde{\zeta }}_{4/\sqrt{\varepsilon }}(x-\beta (u))(|{\nabla u}|^2+|{u}|^2) \in C^1({\widetilde{S}}_{b,\rho _2},{\mathbb {R}}). \end{aligned}$$

(5.1)

We note that $T_\varepsilon (u)$ is translation invariant, that is,

$$\begin{aligned} T_\varepsilon (u(\cdot -h))=T_\varepsilon (u) \quad \text {for all}\ h\in {\mathbb {R}}^N \end{aligned}$$

and

$$\begin{aligned} \Vert {u}\Vert _{H^1(|{x-\beta (u)}|\ge {4\over \sqrt{\varepsilon }})}^2 \le T_\varepsilon (u). \end{aligned}$$

We use $T_\varepsilon (u)$ to estimate the size of u outside of a ball $B(\beta (u),{4\over \sqrt{\varepsilon }})$.

In this section, we extend our idea in [23] to generate a special deformation flow for $I_\varepsilon (u)$, which keeps $T_\varepsilon (u)$ small along the flow.

5.2 A Special Vector Field in $A_{{\rho _{*\!*}}}^{(\varepsilon )}$

To construct a deformation flow which keeps the size of tail $T_\varepsilon (u)$ small, we find a special vector field in this section.

We note $A_{{\rho _{*\!*}}}^{(\varepsilon )}$ is bounded and so there exists $C>0$ such that

$$\begin{aligned} \Vert {u}\Vert _{H^1}^2 \le C \quad \text {for all}\ u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}. \end{aligned}$$

First we decompose $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$ into a center part $u^{(1)}$ and a tail part $u^{(2)}$. We denote the integer part of $a>0$ by [a].

Since

$$\begin{aligned} \sum _{k=0}^{[\varepsilon ^{-1/4}]-1} \Vert {u}\Vert _{H^1(|{x-\beta (u)}|\in [{2\over \sqrt{\varepsilon }}+{k\over \varepsilon ^{1/4}},{2\over \sqrt{\varepsilon }}+{k+1\over \varepsilon ^{1/4}}])}^2 \le \Vert {u}\Vert _{H^1}^2 \le C, \end{aligned}$$

there exists $k\in \{1,2,\cdots , [\varepsilon ^{-1/4}]-1\}$ such that

$$\begin{aligned} \Vert {u}\Vert _{H^1(|{x-\beta (u)}|\in [{2\over \sqrt{\varepsilon }}+{k\over \varepsilon ^{1/4}},{2\over \sqrt{\varepsilon }}+{k+1\over \varepsilon ^{1/4}}])}^2 \le {C\over [\varepsilon ^{-1/4}]} \rightarrow 0 \quad \text {as}\ \varepsilon \rightarrow 0. \end{aligned}$$

(5.2)

In what follows we denote by $c_\varepsilon$ various constants which do not depend on u and satisfy $c_\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$. We set

$$\begin{aligned} u^{(1)}(x) = \zeta _{{2\over \sqrt{\varepsilon }}+{k\over \varepsilon ^{1/4}}}(x-\beta (u)) u(x), \qquad u^{(2)}(x) = {\widetilde{\zeta }}_{{2\over \sqrt{\varepsilon }}+{k+1\over \varepsilon ^{1/4}}}(x-\beta (u)) u(x), \end{aligned}$$

where $\zeta _R(x)$, ${\widetilde{\zeta }}_R(x)$ are defined in (2.1). We also set

$$\begin{aligned} M_1(u) = \zeta _{1/\sqrt{\varepsilon }}(x-\beta (u)) u, \qquad M_2(u) = (1-\zeta _{1/\sqrt{\varepsilon }}(x-\beta (u)))u. \end{aligned}$$

(5.3)

These function also give decomposition of u into a center part and a tail part. Clearly we have $u=M_1(u)+M_2(u)$. By (2.2), we also have $\Vert {M_1(u)}\Vert _{H^1}$, $\Vert {M_2(u)}\Vert _{H^1}\le 3\Vert {u}\Vert _{H^1}$.

We note that $u^{(1)}$, $u^{(2)}$, $M_1(u)$, $M_2(u)$ depend on $\varepsilon$. But for simplicity of notation, we omit $\varepsilon$ from the notation.

We use $-u^{(2)}$ to construct a deformation flow and we use $M_1(u)$ and $M_2(u)$ to estimate effects of $-u^{(2)}$.

$u^{(2)}$ has the following properties.

Lemma 5.1

There exists $c_\varepsilon >0$ independent of $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$ such that

$$\begin{aligned} c_\varepsilon \rightarrow 0 \quad \text {as}\ \varepsilon \rightarrow 0 \end{aligned}$$

and for $\varepsilon >0$ small $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$ satisifes the following properties (i)–(v).

• (i)
  $$\begin{aligned}&\Vert {u^{(2)}}\Vert _{H^1},\ \Vert {M_2(u)}\Vert _{H^1} < \rho _0, \end{aligned}$$

  (5.4)
  $$\begin{aligned}&\Vert {u-u^{(1)}-u^{(2)}}\Vert _{H^1} \le c_\varepsilon , \end{aligned}$$

  (5.5)
  $$\begin{aligned}&|{(u-u^{(2)}, u^{(2)})_{H^1}}| \le c_\varepsilon , \end{aligned}$$

  (5.6)
  $$\begin{aligned}&|{(I_\varepsilon '(u)-I_\varepsilon '(u^{(2)}))u^{(2)}}| \le c_\varepsilon . \end{aligned}$$

  (5.7)

• (ii) For the center of mass $\beta (u)$ defined in Sect. 3.3,

  $$\begin{aligned} \beta '(u) u^{(2)} =0. \end{aligned}$$

  (5.8)

• (iii) For $M_1(u)$, $M_2(u)$ defined in (5.3),

  $$\begin{aligned}&\partial _u M_1(u)u^{(2)} =0, \end{aligned}$$

  (5.9)
  $$\begin{aligned}&\partial _u(\Vert {M_2(u)}\Vert _{H^1}^2)u^{(2)} \ge -c_\varepsilon . \end{aligned}$$

  (5.10)

• (iv) For $T_\varepsilon (u)$ defined in (5.1),

  $$\begin{aligned}&T_\varepsilon (u) \le \Vert {u^{(2)}}\Vert _{H^1}^2, \end{aligned}$$

  (5.11)
  $$\begin{aligned}&T_\varepsilon '(u)u^{(2)} = 2T_\varepsilon (u). \end{aligned}$$

  (5.12)

• (v) For $c_0>0$ given in Corollary 2.2, we have

  $$\begin{aligned} I_\varepsilon '(u)u^{(2)} \ge c_0T_\varepsilon (u) - c_\varepsilon . \end{aligned}$$

  (5.13)

From Lemma 5.1, we can observe a vector field $u\mapsto -u^{(2)}$ has good properties for deformation. By (ii), (iii), $-u^{(2)}$ does not effect the center part $M_1(u)$ and the center $\beta (u)$ of mass of u. By (5.12) and (5.13), $-u^{(2)}$ gives a direction which decreases both of $I_\varepsilon (u)$ and $T_\varepsilon (u)$ provided $T_\varepsilon (u)\ge {c_\varepsilon \over c_0}$. Thus it is convenient to construct a deformation flow for $I_\varepsilon (u)$ which keeps the size $T_\varepsilon (u)$ of tail small.

Proof

• (i) $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}\subset {\widetilde{S}}_{b,{\rho _{*\!*}}}$ can be written as

  $$\begin{aligned} u(x)=\omega (x-p)+\varphi (x), \end{aligned}$$

  where $\omega \in {\widehat{S}}_b$ and $\Vert {\varphi }\Vert _{H^1}<{\rho _{*\!*}}$. Since $|{\beta (u)-p}|\le R_0$ and ${\widehat{S}}_b$ is compact in $H^1({\mathbb {R}}^N)$, we have $\Vert {u}\Vert _{H^1(|{x-\beta (u)}|\ge 1/\sqrt{\varepsilon })} \le 2{\rho _{*\!*}}$ for $\varepsilon$ small. Thus by (2.2)

  $$\begin{aligned} \Vert {u^{(2)}}\Vert _{H^1}, \, \Vert {M_2(u)}\Vert _{H^1} \le 3 \Vert {u}\Vert _{H^1(|{x-\beta (u)}|\ge 1/\sqrt{\varepsilon })} \le 6{\rho _{*\!*}}<\rho _0. \end{aligned}$$

  By (5.2), we have uniformly in $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$,

  $$\begin{aligned} \begin{aligned} \Vert {u-u^{(1)}-u^{(2)}}\Vert _{H^1}&= \Vert { (1-\zeta _{{2\over \sqrt{\varepsilon }}+{k\over \varepsilon ^{1/4}}}(x-\beta (u)) -{\widetilde{\zeta }}_{{2\over \sqrt{\varepsilon }}+{k+1\over \varepsilon ^{1/4}}}(x-\beta (u))) u}\Vert _{H^1} \\&\le 3\Vert {u}\Vert _{H^1(|{x-\beta (u)}|\in [{2\over \sqrt{\varepsilon }}+{k\over \varepsilon ^{1/4}}, {2\over \sqrt{\varepsilon }}+{k+1\over \varepsilon ^{1/4}}])} \rightarrow 0 \quad \text {as}\ \varepsilon \rightarrow 0. \end{aligned} \end{aligned}$$

  We also have

  $$\begin{aligned} |{(u-u^{(2)}, u^{(2)})_{H^1}}| \le C\Vert {u}\Vert _{H^1(|{x-\beta (u)}|\in [{2\over \sqrt{\varepsilon }}+{k\over \varepsilon ^{1/4}}, {2\over \sqrt{\varepsilon }}+{k+1\over \varepsilon ^{1/4}}])}^2 \rightarrow 0 \quad \text {as}\ \varepsilon \rightarrow 0. \end{aligned}$$

  Thus we have (5.5) and (5.6). In a similar way, using Lemma 2.3 (ii) with $R={2\over \sqrt{\varepsilon }}+{k\over \varepsilon ^{1/4}}$ and $L={1\over \varepsilon ^{1/4}}$, we have (5.7).

• (ii) Since $\mathop {\textrm{supp}}u^{(2)}\subset {\mathbb {R}}^N{\setminus } B(\beta (u),{2\over \sqrt{\varepsilon }})$ does not intersect $B(\beta (u), 3R_0)$ for $\varepsilon \in (0,{1\over 9R_0^2})$, we have by (iii) of Proposition 3.3

  $$\begin{aligned} \beta (u+ t\ell u^{(2)}) =\beta (u) \quad \text {for small}\ t. \end{aligned}$$

  Thus we have (5.8).

  By $\mathop {\textrm{supp}}u^{(2)}\subset {\mathbb {R}}^N{\setminus } B(\beta (u),{2\over \sqrt{\varepsilon }})$ we note that

  $$\begin{aligned} \zeta _{1/\sqrt{\varepsilon }}(x-\beta (u))u^{(2)}(x)=0. \end{aligned}$$

  (5.14)

• (iii) We have from (5.8), (5.14)

  $$\begin{aligned} \partial _u M_1(u)u^{(2)} = -\zeta _{1/\sqrt{\varepsilon }}'(x-\beta (u))(\beta '(u)u^{(2)}) u +\zeta _{1/\sqrt{\varepsilon }}(x-\beta (u)) u^{(2)} = 0. \end{aligned}$$

  Thus we have (5.9).

  For $M_2(u)$, we compute by (5.6)

  $$\begin{aligned} \begin{aligned} {1 \over 2}\partial _u&(\Vert {M_2(u)}\Vert _{H^1}^2)u^{(2)} = (M_2(u), \partial _u M_2(u)u^{(2)} )_{H^1} \\ =&( (1-\zeta _{1/\sqrt{\varepsilon }}(x-\beta (u)) u(x), \zeta _{1/\sqrt{\varepsilon }}'(x-\beta (u))(\beta '(u)u^{(2)}) u \\&+(1-\zeta _{1/\sqrt{\varepsilon }})(x-\beta (u))u^{(2)})_{H^1} \\ =&(u, u^{(2)})_{H^1} =\Vert {u^{(2)}}\Vert _{H^1}^2 + (u-u^{(2)}, u^{(2)})_{H^1} \ge (u-u^{(2)}, u^{(2)})_{H^1} \\ \ge&-c_\varepsilon . \end{aligned} \end{aligned}$$

  Thus we have (5.10).

• (iv) Since $u(x)=u^{(2)}(x)$ in $\mathop {\textrm{supp}}{\widetilde{\zeta }}_{4/\sqrt{\varepsilon }}(x-\beta (u))= {\mathbb {R}}^N{\setminus } B(\beta (u),{4\over \sqrt{\varepsilon }}-1)$, we have (5.11) and

  $$\begin{aligned} \begin{aligned} T_\varepsilon '(u)u^{(2)} =&-\int _{{\mathbb {R}}^N}{\widetilde{\zeta }}_{4/\sqrt{\varepsilon }}'(x-\beta (u)) (\beta '(u)u^{(2)}) (|{\nabla u}|^2+u^2) \\&+ 2\int _{{\mathbb {R}}^N}{\widetilde{\zeta }}_{4/\sqrt{\varepsilon }}(x-\beta (u)) ( \nabla u\nabla u^{(2)}+u u^{(2)} ) \\ =&2T_\varepsilon (u). \end{aligned} \end{aligned}$$

  Thus we have (5.12).

• (v) By (5.4), (5.7), (5.11) and Corollary 2.2,

  $$\begin{aligned} I_\varepsilon '(u)u^{(2)} \ge I_\varepsilon '(u^{(2)}) u^{(2)}-c_\varepsilon \ge c_0\Vert {u^{(2)}}\Vert _{H^1}^2-c_\varepsilon \ge c_0 T_\varepsilon (u)-c_\varepsilon . \end{aligned}$$

  Thus we get (v).

$\square$

Choice of $\kappa _\varepsilon$. By the compactness of ${\widehat{S}}_b$, we have

$$\begin{aligned} \sup _{\omega \in {\widehat{S}}_b} T_\varepsilon (\omega ) \rightarrow 0 \quad \text {as} \ \varepsilon \rightarrow 0. \end{aligned}$$

For $c_\varepsilon >0$ given in Lemma 5.1, we set

$$\begin{aligned} \kappa _\varepsilon \equiv \max \left\{ 2\sup _{\omega \in {\widehat{S}}_b} T_\varepsilon (\omega ), {2c_\varepsilon \over c_0}\right\} \rightarrow 0 \quad \text {as}\ \varepsilon \rightarrow 0. \end{aligned}$$

(5.15)

With this choice of $\kappa _\varepsilon$, we have the following corollary. In what follows, we use the following notation for $c\in {\mathbb {R}}$

$$\begin{aligned} \begin{aligned}&[I_\varepsilon \le c] =\{ u\in H^1({\mathbb {R}}^N):\, I_\varepsilon (u)\le c\}, \\&[T_\varepsilon \ge c] =\{ u\in H^1({\mathbb {R}}^N):\, T_\varepsilon (u)\ge c\}. \end{aligned} \end{aligned}$$

Corollary 5.2

For $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}\cap [T_\varepsilon \ge \kappa _\varepsilon ]$, we have

$$\begin{aligned} I_\varepsilon '(u)u^{(2)} \ge c_\varepsilon , \end{aligned}$$

in particular, $I_\varepsilon '(u)\not =0$ in $A_{{\rho _{*\!*}}}^{(\varepsilon )}\cap [T_\varepsilon \ge \kappa _\varepsilon ]$.

Proof

By (v) of Lemma 5.1, we have for $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}\cap [T_\varepsilon \ge \kappa _\varepsilon ]$.

$$\begin{aligned} I_\varepsilon '(u)u^{(2)} \ge c_0T_\varepsilon (u) -c_\varepsilon \ge c_0\kappa _\varepsilon -c_\varepsilon \ge c_0\cdot {2c_\varepsilon \over c_0}-c_\varepsilon = c_\varepsilon . \end{aligned}$$

$\square$

As a corollary to Proposition 4.9 (ii) and Corollary 5.2, we have

Corollary 5.3

Suppose that for $\varepsilon >0$

$$\begin{aligned} (H_\varepsilon (u),I_\varepsilon '(u))\not =(0,0) \quad \text {for}\ u\in \overline{N_{{\rho _{*}}}^{(\varepsilon )}}\cap [T_\varepsilon \le \kappa _\varepsilon ] \ \text {with}\ I_\varepsilon (u)\in [b-\delta _0,b+\delta _0].\nonumber \\ \end{aligned}$$

(5.16)

Then there exists $\nu _\varepsilon >0$ such that

$$\begin{aligned} \Vert {(H_\varepsilon (u),I_\varepsilon '(u))}\Vert _{({{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)})^*} \ge \nu _\varepsilon \end{aligned}$$

(5.17)

for $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$ with $I_\varepsilon (u)\in [b-\delta _0,b+\delta _0]$.

In fact, Corollary 5.2 and (5.16) imply (4.22). Thus Proposition 4.9 (ii) implies (5.17).

For later use, we state the following lemma, which states that the property $u\in [T_\varepsilon \le \kappa _\varepsilon ]$ ensures that u concentrates around the center of mass $\beta (u)$.

Proposition 5.4

Assume $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}\cap [T_\varepsilon \le \kappa _\varepsilon ]$. Then we have

$$\begin{aligned} I_\varepsilon (u) \ge L(\varepsilon \beta (u),u) -c_\varepsilon -{1 \over 2}{\overline{V}}\kappa _\varepsilon . \end{aligned}$$

Here $c_\varepsilon >0$ is independent of u and satisfies $c_\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$.

Proof

For $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}\cap [T_\varepsilon \le \kappa _\varepsilon ]$ we compute

$$\begin{aligned} \begin{aligned} I_\varepsilon (u)&= L(\varepsilon \beta (u),u) +{1 \over 2}\int _{{\mathbb {R}}^N}(V(\varepsilon x)-V(\varepsilon \beta (u)))u^2 \\&= L(\varepsilon \beta (u),u) +{1 \over 2}\left( \int _{|{x-\beta (u)}|\le {4\over \sqrt{\varepsilon }}} + \int _{|{x-\beta (u)}|\ge {4\over \sqrt{\varepsilon }}} \right) (V(\varepsilon x)-V(\varepsilon \beta (u)))u^2 \\&\ge L(\varepsilon \beta (u),u) -{1 \over 2}\Vert {V(y)-V(\varepsilon \beta (u))}\Vert _{L^\infty (|{y-\varepsilon \beta (u)}|\le {4\sqrt{\varepsilon }})} \Vert {u}\Vert _2^2\\&\quad -{1 \over 2}{\overline{V}}\Vert {u}\Vert _{H^1(|{x-\beta (u)}|\ge {4\over \sqrt{\varepsilon }})}^2 \\&\ge L(\varepsilon \beta (u),u) -{1 \over 2}\Vert {V(y)-V(\varepsilon \beta (u))}\Vert _{L^\infty (|{y-\varepsilon \beta (u)}|\le {4\sqrt{\varepsilon }})} \Vert {u}\Vert _2^2 -{1 \over 2}{\overline{V}} T_\varepsilon (u). \end{aligned} \end{aligned}$$

Let $\Omega _{\varepsilon R_0+{\rho _{*\!*}}}$ be a $(\varepsilon R_0+{\rho _{*\!*}})$-neighborhood of $\Omega$, that is, $\Omega _{\varepsilon R_0+{\rho _{*\!*}}} = \{ x\in {\mathbb {R}}^N:\, \mathop {\textrm{dist}}\nolimits _{{\mathbb {R}}^N}(x,\Omega )\le \varepsilon R_0+{\rho _{*\!*}}\}$. We recall (4.21) and we note that $A_{{\rho _{*\!*}}}^{(\varepsilon )}$ is uniformly bounded for all $\varepsilon \in (0,1]$ and let $C=\sup _{\varepsilon \in (0,1], u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}}\Vert {u}\Vert _2^2<\infty$. Setting

$$\begin{aligned} c_\varepsilon = {1 \over 2}C\sup \{ |{V(y)-V(y')}|:\, y, y'\in \Omega _{\varepsilon R_0+{\rho _{*\!*}}}, \, |{y-y'}|\le 4\sqrt{\varepsilon }\} \rightarrow 0 \quad \text {as}\ \varepsilon \rightarrow 0, \end{aligned}$$

and noting $u\in [T_\varepsilon \le \kappa _\varepsilon ]$, we have the conclusion of Proposition 5.4. $\square$

6 Deformation Argument

6.1 Deformation Result

In this section we develop a special deformation argument for $I_\varepsilon (u)$, which keeps $T_\varepsilon (u)$ small. Our aim is to show the following deformation result.

Proposition 6.1

Let $\varepsilon _0$, $\nu _0$, $\delta _0>0$ be numbers given in Proposition 4.9 and let $\kappa _\varepsilon >0$ be a number given in (5.15), which satisfies $\kappa _\varepsilon \rightarrow 0$ as $\varepsilon \rightarrow 0$. Moreover suppose for some $\varepsilon \in (0,\varepsilon _0]$

$$\begin{aligned} (H_\varepsilon (u),I_\varepsilon '(u)) \not = 0 \quad \text {for}\ u\in \overline{N_{{\rho _{*}}}^{(\varepsilon )}}\ \text {with} \ I_\varepsilon (u)\in [b-\delta _0,b+\delta _0]. \end{aligned}$$

(6.1)

Then for any $\delta _1\in (0,\delta _0)$ there exist $\delta \in (0,\delta _1)$ and a continuous map $\eta (t,u):\, [0,1]\times A_{{\rho _{*\!*}}}^{(\varepsilon )}\rightarrow A_{{\rho _{*\!*}}}^{(\varepsilon )}$ such that

• (i) $\eta (0,u)=u$ for all $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$.

• (ii) $\eta (t,u)=u$ for all $t\in [0,1]$ if $I_\varepsilon (u)\not \in [b-\delta _1,b+\delta _1]$ or $u\not \in A_{3{\rho _{*\!*}}+{\rho _{*}}\over 4}^{(\varepsilon )}$.

• (iii) $t\mapsto I_\varepsilon (\eta (t,u))$ is a non-increasing function of t for all $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$.

• (iv) $\eta (1,u)\in [I_\varepsilon \le b-\delta ]$ if $u\in A_{{\rho _{*}}}^{(\varepsilon )}\cap [I_\varepsilon \le b+\delta ]$.

• (v) $\eta (t,u)\in [T_\varepsilon \le \kappa _\varepsilon ]$ for all $t\in [0,1]$ if $u\in [T_\varepsilon \le \kappa _\varepsilon ]$.

The properties (i)–(iv) are standard under the standard Palais-Smale condition. However our concentration-compactness type result Proposition 4.5 ensures a weaker condition; we assume (4.10) in addition to (4.8) and (4.9) and we have (4.23) under the condition (4.22).

We note that $H_\varepsilon (u)$ gives a useful information on deformation. In fact, for $h\in {\mathbb {R}}^N$ we have

$$\begin{aligned} \begin{aligned} {d\over dt}\Big |_{t=0} I_\varepsilon (u(x-{h\over \varepsilon }t))&= {1 \over 2}{d\over dt}\Big |_{t=0} \int _{{\mathbb {R}}^N}V(\varepsilon x+ht)u(x)^2 = {1 \over 2}\int _{{\mathbb {R}}^N}h\cdot \nabla V(\varepsilon x)u(x)^2 \\&= {1 \over 2}h\cdot H_\varepsilon (u). \end{aligned} \end{aligned}$$

Thus, if $H_\varepsilon (u)\not =0$, the translation flow:

$$\begin{aligned} (t,v)\mapsto v\left( x-{h\over \varepsilon } t\right) \quad \text {with}\ h=-H_\varepsilon (u) \end{aligned}$$

(6.2)

gives a decreasing flow in a neighborhood of u.

The property (v) means that the set $[T_\varepsilon \le \kappa _\varepsilon ]$ is positively invariant for the flow $\eta (t,u)$, i.e.,

$$\begin{aligned} \eta (t,[T_\varepsilon \le \kappa _\varepsilon ])\subset [T_\varepsilon \le \kappa _\varepsilon ] \quad \text {for}\ t\ge 0. \end{aligned}$$

(6.3)

This property is related to the tail minimizing flow developed in [23]. In [23], we used the tail minimizing flow separately from the deformation flow (the steepest descent flow) for $I_\varepsilon (u)$. Here, extending the idea in [23] we construct a deformation flow for $I_\varepsilon (u)$ which keeps the size $T_\varepsilon (u)$ of the tail $u|_{{\mathbb {R}}^N\setminus B(\beta (u),4/\sqrt{\varepsilon })}$ small.

Remark 6.2

In [25, 33, 35], we study radially symmetric problems in ${\mathbb {R}}^N$. A typical example is a nonlinear scalar field equation: $-\Delta u = g(u)$ in ${\mathbb {R}}^N$. The natural corresponding functional is

$$\begin{aligned} \mathcal{I}(u)={1 \over 2}\Vert {\nabla u}\Vert _2^2 -\int _{{\mathbb {R}}^N}G(u):\, H_r^1({\mathbb {R}}^N)\rightarrow {\mathbb {R}}\end{aligned}$$

and scaling $\theta \mapsto u(x/e^\theta )$ is important in the arguments in [25, 33, 35]. Precisely Pohozaev functional

$$\begin{aligned} \mathcal{P}(u)={N-2\over 2}\Vert {\nabla u}\Vert _2^2 -N\int _{{\mathbb {R}}^N}G(u) \end{aligned}$$

is characterized as

$$\begin{aligned} \mathcal{P}(u)={d\over d\theta }\Bigl |_{\theta =0} \mathcal{I}(u(x/e^\theta )). \end{aligned}$$

(6.4)

Thus, if $\mathcal{P}(u)>0$ ($\mathcal{P}(u)<0$ resp.), the scaling flow $(\theta ,u)\mapsto u(x/e^{-\theta })$ ($u(x/e^\theta )$ resp.) gives a decreasing flow in a neighborhood of u. In [25, 33, 35], we introduce an augmented functional $\mathcal{J}(\theta ,u)$ by

$$\begin{aligned} \mathcal{J}(\theta ,u)={1 \over 2}e^{(N-2)\theta }\Vert {\nabla u}\Vert _2^2 -e^{N\theta } \int _{{\mathbb {R}}^N}G(u), \end{aligned}$$

which enjoys the property $\mathcal{J}(\theta ,u)=\mathcal{I}(u(x/e^\theta ))$. We develop a deformation flow for $\mathcal{I}(u)$ through a deformation for $\mathcal{J}(\theta ,u)$ in the augmented space ${\mathbb {R}}\times H_r^1({\mathbb {R}}^N)$.

In the following sections, replacing scaling (6.4) to translation (6.2), we give a proof of Proposition 6.1.

6.2 Augmented Functional

To prove Proposition 6.1, we consider the following functional in the augmented space ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$:

$$\begin{aligned} J_\varepsilon (z,u) = {1 \over 2}\Vert {\nabla u}\Vert _2^2 +{1 \over 2}\int _{{\mathbb {R}}^N}V(\varepsilon x+z)u(x)^2 -{1 \over 2}\mathcal{D}(u):\, {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}\rightarrow {\mathbb {R}}. \end{aligned}$$

We note that $J_\varepsilon (z,u)\in C^1({{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)},{\mathbb {R}})$ and

• (i) $J_\varepsilon (z,u)=I_\varepsilon (u(x-{z\over \varepsilon }))$.

• (ii) $\partial _uJ_\varepsilon (z,u)\varphi =I_\varepsilon '(u(x-{z\over \varepsilon }))\varphi (x-{z\over \varepsilon })$.

• (iii) $\partial _zJ_\varepsilon (z,u)=H_\varepsilon (u(x-{z\over \varepsilon }))$.

Recalling $D=(\partial _z,\partial _u)$, we have

Lemma 6.3

• (i) For $(z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$, (z, u) is a critical point of $J_\varepsilon$, i.e., $DJ_\varepsilon (z,u)=0$ if and only if $v(x)=u(x-{z\over \varepsilon })$ satisfies

  $$\begin{aligned} I_\varepsilon '(v)=0 \quad \text {and} \quad H_\varepsilon (v)=0. \end{aligned}$$

• (ii) For $c\in {\mathbb {R}}$, c is a critical value of $J_\varepsilon$ if and only if there exists $v\in H^1({\mathbb {R}}^N)$ such that

  $$\begin{aligned} I_\varepsilon (v)=c, \quad I_\varepsilon '(v)=0 \quad \text {and} \quad H_\varepsilon (v)=0. \end{aligned}$$

• (iii) For all $(z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$

  $$\begin{aligned} \Vert {DJ_\varepsilon (z,u)}\Vert _{({{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)})^*}^2 = {\left| H_\varepsilon \left( u\left( x-{z\over \varepsilon }\right) \right) \right| }^2 + {\left\| I_\varepsilon '\left( u\left( x-{z\over \varepsilon }\right) \right) \right\| }_{(H^1({\mathbb {R}}^N))^*}^2. \end{aligned}$$

As in Corollary 2.2, we have

Corollary 6.4

There exist $\rho _0>0$ and $c_0>0$ such that

$$\begin{aligned} J_\varepsilon (z,u)\ge & {} c_0\Vert {u}\Vert _{H^1}^2, \quad \partial _uJ_\varepsilon (z,u)u \ge c_0\Vert {u}\Vert _{H^1}^2 \end{aligned}$$

for all $(z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ with $\Vert {u}\Vert _{H^1} \le \rho _0$.

To show our Proposition 6.1, we develop a deformation argument in ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ and we construct a flow $\eta (t,u)$ through a flow ${\widetilde{\eta }}(t,z,u)$ on a product space ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$.

We introduce a pseudo-distance $\mathop {\textrm{DIST}}\nolimits _\varepsilon (\cdot ,\cdot )$ on ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$, which is related to $\mathop {\textrm{dist}}\nolimits _\varepsilon (\cdot ,\cdot )$, by

$$\begin{aligned} \mathop {\textrm{DIST}}\nolimits _\varepsilon ((z,u),(z',u')) = \inf _{h\in {\mathbb {R}}^N} \sqrt{|{z'-z-h}|^2 +{\left\| u\left( x-{h\over \varepsilon }\right) -u'(x)\right\| }_{H^1}^2} \end{aligned}$$

for (z, u), $(z',u')\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$. We note that

$$\begin{aligned} \mathop {\textrm{DIST}}\nolimits _\varepsilon ((z,u),(z',u')) = \mathop {\textrm{dist}}\nolimits _\varepsilon \left( u\left( x-{z\over \varepsilon }\right) ,u'\left( x-{z'\over \varepsilon }\right) \right) \end{aligned}$$

and

$$\begin{aligned} \mathop {\textrm{DIST}}\nolimits _\varepsilon ((z,u),(z',u'))\le & {} \mathop {\textrm{dist}}\nolimits _{{{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}}((z,u), (z',u')) \\\equiv & {} \sqrt{|{z-z'}|^2+\Vert {u-u'}\Vert _{H^1}^2}. \end{aligned}$$

We set

$$\begin{aligned} \begin{aligned} \mathcal{N}_\rho ^{(\varepsilon )}&= \{ (z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}:\, \mathop {\textrm{DIST}}\nolimits _\varepsilon ((z,u),\widehat{\mathcal{K}}_b)<\rho \} \\&= \left\{ (z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}:\, \mathop {\textrm{dist}}\nolimits _\varepsilon \left( u\left( x-{z\over \varepsilon }\right) ,{\widehat{K}}_b^{(\varepsilon )}\right) )<\rho \right\} , \\ \mathcal{A}_\rho ^{(\varepsilon )}&= \{ (z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}:\, \mathop {\textrm{DIST}}\nolimits _\varepsilon ((z,u),\widehat{\mathcal{Z}}_b)<\rho \} \\&= \left\{ (z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}:\, \mathop {\textrm{dist}}\nolimits _\varepsilon \left( u\left( x-{z\over \varepsilon }\right) , {\widehat{Z}}_b^{(\varepsilon )}\right) <\rho \right\} . \end{aligned} \end{aligned}$$

Clearly these sets are uniformly bounded with respect to $\varepsilon \in (0,1]$ and we have $\mathcal{N}_\rho ^{(\varepsilon )} \subset \mathcal{A}_\rho ^{(\varepsilon )}.$ From Proposition 4.9 (i), Corollary 5.3 and Lemma 6.3 we have the following

Proposition 6.5

Let $0<{\rho _{*}}<{\rho _{*\!*}}$ be the numbers satisfying (4.20). Then we have

• (i) There exist $\nu _0>0$ and $\delta _0>0$ independent of $\varepsilon$ such that for $\varepsilon >0$ small

  $$\begin{aligned} \Vert {DJ_\varepsilon (z,u)}\Vert _{({{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)})^*} \ge \nu _0 \end{aligned}$$

  (6.5)

  for all $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}{\setminus } \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}$ with $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$.

• (ii) Suppose that (6.1) holds, in other words, it holds that

  $$\begin{aligned} DJ_\varepsilon (z,u)\not =0 \quad \text {for all} \ (z,u)\in \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}\ \text {with} \ J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]. \end{aligned}$$

  (6.6)

  Then there exists $\nu _\varepsilon >0$ such that

  $$\begin{aligned}{} & {} \Vert {DJ_\varepsilon (z,u)}\Vert _{({{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)})^*} \ge \nu _\varepsilon \quad \text {for all} \ (z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\ \text {with} \nonumber \\{} & {} \quad J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]. \end{aligned}$$

  (6.7)

We note that we may assume $\nu _\varepsilon < \nu _0$.

6.3 Construction of a Vector Field

In what follows, we will show that the existence of a critical point $(z,u)\in \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}$ with $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$. Arguing indirectly, we assume (6.1) holds. To construct a deformation flow, we find a special vector field $V_{z,u}:\, \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\rightarrow {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$. Since (6.5) and (6.7) hold by Proposition 6.5, for $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ with $J_\varepsilon (z,u) \in [b-\delta _0,b+\delta _0]$ there exists $(\xi ,w)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ such that

$$\begin{aligned}&|{\xi }|^2 +\Vert {w}\Vert _{H^1}^2 \le 1, \end{aligned}$$

(6.8)

$$\begin{aligned}&DJ_\varepsilon (z,u)(\xi ,w) > \nu _0 \quad \text {if}\ (z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\setminus \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}, \end{aligned}$$

(6.9)

$$\begin{aligned}&DJ_\varepsilon (z,u)(\xi ,w) > \nu _\varepsilon \quad \text {if}\ (z,u)\in \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}. \end{aligned}$$

(6.10)

We compute for $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ and $\ell \ge 0$

$$\begin{aligned} \partial _u T_\varepsilon (u)(w+\ell u^{(2)}) = \partial _u T_\varepsilon (u)w +\ell \partial _u T_\varepsilon (u)u^{(2)} \ge -C_1 + 2\ell T_\varepsilon (u), \end{aligned}$$

(6.11)

where $C_1>0$ is independent of $\varepsilon$ and u. Here we used (5.12) and the boundedness of $\Vert {\partial _uT_\varepsilon (u)}\Vert _{H^1({\mathbb {R}}^N)^*}$.

For $\kappa _\varepsilon$ defined in (5.15), we set

$$\begin{aligned} \ell _\varepsilon \equiv {C_1\over \kappa _\varepsilon } \rightarrow \infty \quad \text {as}\ \varepsilon \rightarrow 0. \end{aligned}$$

(6.12)

Finally we define $V_{z,u}\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ for $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ with $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$ by

$$\begin{aligned} V_{z,u} = {\left\{ \begin{array}{ll} (\xi ,w+\ell _\varepsilon u^{(2)}) &{}\text {if}\ T_\varepsilon (u)\ge \kappa _\varepsilon ,\\ (\xi ,w) &{}\text {if}\ T_\varepsilon (u)<\kappa _\varepsilon . \end{array}\right. } \end{aligned}$$

Then we have

Proposition 6.6

Suppose that (6.1) holds. Then for $\varepsilon \in (0,{1\over 9R_0^2})$ and $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$, we have

• (i) If $T_\varepsilon (u)\ge \kappa _\varepsilon$, then

  $$\begin{aligned} DT_\varepsilon (u) V_{z,u} >0. \end{aligned}$$

• (ii) For $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ with $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$,

  $$\begin{aligned} DJ_\varepsilon (z,u) V_{z,u} > \nu _\varepsilon . \end{aligned}$$

• (iii) For $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}{\setminus } \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}$ with $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$,

  $$\begin{aligned} DJ_\varepsilon (z,u) V_{z,u} > \nu _0. \end{aligned}$$

• (iv) There exist C, $C'>0$ such that for $M_1(u)$, $M_2(u)$ given in (5.3)

  $$\begin{aligned}&\Vert {DM_1(u)V_{z,u}}\Vert _{H^1} < C, \end{aligned}$$

  (6.13)
  $$\begin{aligned}&D(\Vert {M_2(u)}\Vert _{H^1}^2) V_{z,u} > -C'. \end{aligned}$$

  (6.14)

In the above proposition, we write

$$\begin{aligned} DT_\varepsilon (u)=(0,\partial _uT_\varepsilon (u)), \quad DM_i(u)=(0,\partial _u M_i(u)) \quad \text {for}\ i=1,2. \end{aligned}$$

In particular,

$$\begin{aligned} DT_\varepsilon (u)V_{z,u} ={\left\{ \begin{array}{ll} \partial _u T_\varepsilon (u)(w+\ell _\varepsilon u^{(2)}) &{}\text {if}\ T_\varepsilon (u)\ge \kappa _\varepsilon ,\\ \partial _u T_\varepsilon (u)w &{}\text {if}\ T_\varepsilon (u)<\kappa _\varepsilon . \end{array}\right. } \end{aligned}$$

We use similar formulas also for $M_1(u)$ and $\Vert {M_2(u)}\Vert _{H^1}^2$.

Proof

First we recall that (6.5), (6.7) hold under (6.1).

(i) By (6.11) and (6.12), we have for $T_\varepsilon (u)\ge \kappa _\varepsilon$

$$\begin{aligned} DT_\varepsilon (u)V_{z,u} \ge -C_1 +2\ell _\varepsilon T_\varepsilon (u) \ge -C_1+ 2\ell _\varepsilon \kappa _\varepsilon = C_1>0. \end{aligned}$$

Thus we have (i).

(ii), (iii) By our choice (5.15) of $\kappa _\varepsilon$, as in Corollary 5.2 we have $DJ_\varepsilon (z,u)(0,u^{(2)}) \ge 0$ when $T_\varepsilon (u)\ge \kappa _\varepsilon$. Thus (ii) and (iii) follow from (6.9)–(6.10).

(iv) Since

$$\begin{aligned} \partial _u M_1(u)w = -\zeta _{1/\sqrt{\varepsilon }}'(x-\beta (u))(\beta '(u)w) u +\zeta _{1/\sqrt{\varepsilon }}(x-\beta (u)) w, \end{aligned}$$

$\Vert {\partial _u M_1(u)w}\Vert _{H^1}$ and $\Vert {\partial _u M_2(u)w}\Vert _{H^1}$ are uniformly bounded by the boundedness of $\Vert {\beta '(u)}\Vert _{H^1({\mathbb {R}}^N)^*}$. Thus (6.13) follows from (5.9). As to (6.14), we have from (5.10) and (6.12)

$$\begin{aligned} \partial _u(\Vert {M_2(u)}\Vert _{H^1}^2)\ell _\varepsilon u^{(2)} \ge -\ell _\varepsilon c_\varepsilon = -C_1. \end{aligned}$$

Thus (6.14) follows from the boundedness of $\Vert {\partial _u M_2(u)w}\Vert _{H^1}$. $\square$

Proposition 6.7

Suppose that (6.1) holds. Then for $\varepsilon >0$ small, there exists a locally Lipschitz vector field $W(z,u):\,\mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\cap \{(z,u):\, J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]\} \rightarrow {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ with the following properties.

• (i) $DT_\varepsilon (u)W(z,u)> 0$ if $T_\varepsilon (u) > \kappa _\varepsilon$.

• (ii) $DJ_\varepsilon (z,u)W(z,u) > \nu _\varepsilon$ if $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ and $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$.

• (iii) $DJ_\varepsilon (z,u)W(z,u) > \nu _0$ if $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\setminus \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}$ and $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$.

• (iv) $\Vert {DM_1(u)W(z,u)}\Vert _{H^1} \le C$, $D(\Vert {M_2(u)}\Vert _{H^1}^2)W(z,u)\ge -C'$.

Proof

Let $V_{z,u}$ be a vector field given in Proposition 6.6. We remark that for any $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ there exists a small neighborhood $U_{z,u}$ of (z, u) in ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ such that for $(z',u')\in U_{z,u}$

• (i) $DT_\varepsilon (u')V_{z,u}>0$ if $T_\varepsilon (u)>\kappa _\varepsilon$.

• (ii) $DJ_\varepsilon (z',u')V_{z,u}>\nu _\varepsilon$ if $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ and $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$.

• (iii) $DJ_\varepsilon (z',u')V_{z,u}>\nu _0$ if $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\setminus \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}$ and $J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]$.

• (iv) $\Vert {DM_1(u')V_{z,u}}\Vert _{H^1}<C$, $D(\Vert {M_2(u')}\Vert _{H^1}^2)V_{z,u}>-C'$.

We may choose a neighborhood $U_{z,u}$ of (z, u) so that

$$\begin{aligned} \begin{aligned}&U_{z,u} \subset \{ (z',u'):\, T_\varepsilon (u')>\kappa _\varepsilon \} \quad \text {if}\ \ T_\varepsilon (u)>\kappa _\varepsilon , \\&U_{z,u} \subset \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\setminus \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}\quad \text {if}\ \ (z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\setminus \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}. \end{aligned} \end{aligned}$$

Using a partition of unity, we can construct a locally Lipschitz continuous vector field $W(z,u):\, \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\cap \{(z,u):\, J_\varepsilon (z,u)\in [b-\delta _0,b+\delta _0]\} \rightarrow {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$ in a standard way. We can easily see that W(z, u) satisfies (i)–(iv). $\square$

We note that W(z, u) is bounded in the following sense:

$$\begin{aligned} \Vert {W(z,u)}\Vert _{{{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}} \le C(1+\ell _\varepsilon ) \end{aligned}$$

(6.15)

for all (z, u), where $C>0$ is independent of $\varepsilon$, (z, u).

6.4 Deformation Flow for the Augmented Functional $J_\varepsilon (z,u)$

Using the pseudo-gradient flow W(z, u) obtained in Proposition 6.7, we have

Proposition 6.8

For $\varepsilon >0$ small, suppose that (6.1) holds. Then for any given $\delta _1\in (0,\delta _0)$ there exist $\delta \in (0,\delta _1)$ and a continuous map ${\widetilde{\eta }}(t,z,u):\, [0,1]\times \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}\rightarrow \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ such that

• (i) ${\widetilde{\eta }}(0,z,u)=(z,u)$ for all $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$.

• (ii) ${\widetilde{\eta }}(t,z,u)=(z,u)$ for all $t\in [0,1]$ if $J_\varepsilon (z,u)\not \in [b-\delta _1,b+\delta _1]$ or $(z,u)\not \in \mathcal{A}_{3{\rho _{*\!*}}+{\rho _{*}}\over 4}^{(\varepsilon )}$.

• (iii) $t\mapsto J_\varepsilon ({\widetilde{\eta }}(t,z,u))$ is non-increasing on [0, 1] for all $(z,u)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$.

• (iv) $J_\varepsilon ({\widetilde{\eta }}(1,z,u)) \le b-\delta$ if $(z,u)\in \mathcal{A}_{{\rho _{*}}}^{(\varepsilon )}$ satisfies $J_\varepsilon (z,u)\le b+\delta$.

• (v) $T_\varepsilon ({\widetilde{\eta }}(1,z,u))\le \kappa _\varepsilon$ if $T_\varepsilon (u)\le \kappa _\varepsilon$.

For a proof we use notation for $c\in {\mathbb {R}}$

$$\begin{aligned}{}[[J_\varepsilon \le c]] =\{(z,u)\in {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}:\, J_\varepsilon (z,u)\le c\}. \end{aligned}$$

Proof

Let W(z, u) be a locally Lipschitz continuous vector field given in Proposition 6.7. For $\delta \in (0,{1 \over 2}\delta _1)$ we choose locally Lipschitz continuous functions $\varphi _1:\, {\mathbb {R}}\rightarrow [0,1]$, $\varphi _2:\,{{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}\rightarrow [0,1]$ such that

$$\begin{aligned} \varphi _1(s)={\left\{ \begin{array}{ll} 1 &{}\text {for}\ s\in [b-\delta , b+\delta ],\\ 0 &{}\text {for}\ s\not \in [b-2\delta , b+2\delta ], \end{array}\right. } \qquad \varphi _2(z,u) = {\left\{ \begin{array}{ll} 1 &{}\text {for}\ (z,u)\in \mathcal{A}_{{\rho _{*\!*}}+{\rho _{*}}\over 2}^{(\varepsilon )},\\ 0 &{}\text {for}\ (z,u)\not \in \mathcal{A}_{3{\rho _{*\!*}}+{\rho _{*}}\over 4}^{(\varepsilon )}. \end{array}\right. } \end{aligned}$$

We consider the following ODE:

$$\begin{aligned} {d{\widetilde{\eta }}\over dt} = -\varphi _1(J_\varepsilon ({\widetilde{\eta }}))\varphi _2({\widetilde{\eta }})W({\widetilde{\eta }}), \qquad {\widetilde{\eta }}(0,z,u) = (z,u). \end{aligned}$$

(6.16)

First we note that for each $\varepsilon \in (0,1]$ the vector field W(z, u) is locally Lipschitz and uniformly bounded, where the bound depends on $\varepsilon$ (c.f. (6.15)), the solution ${\widetilde{\eta }}(t)={\widetilde{\eta }}(t,z,u)$ of (6.16) is extendable as long as ${\widetilde{\eta }}(t)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$. Moreover the right hand side of (6.16) vanishes in $\mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}{\setminus } \mathcal{A}_{3{\rho _{*\!*}}+{\rho _{*}}\over 4}^{(\varepsilon )}$ and thus ${\widetilde{\eta }}(t)$ exists for all $t\ge 0$.

We compute

$$\begin{aligned} \begin{aligned} {d\over dt}J_\varepsilon ({\widetilde{\eta }})&= DJ_\varepsilon ({\widetilde{\eta }}){d{\widetilde{\eta }}\over dt} = -\varphi _1(J_\varepsilon ({\widetilde{\eta }}))\varphi _2({\widetilde{\eta }})DJ_\varepsilon ({\widetilde{\eta }})W({\widetilde{\eta }}), \\ {d\over dt}T_\varepsilon ({\widetilde{\eta }})&= -\varphi _1(J_\varepsilon ({\widetilde{\eta }}))\varphi _2({\widetilde{\eta }})DT_\varepsilon ({\widetilde{\eta }})W({\widetilde{\eta }}). \end{aligned} \end{aligned}$$

Thus, we have from Proposition 6.7 that

$$\begin{aligned} {d\over dt}J_\varepsilon ({\widetilde{\eta }})&\le 0 \quad \text {on}\ \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}, \end{aligned}$$

(6.17)

$$\begin{aligned} {d\over dt}J_\varepsilon ({\widetilde{\eta }})&\le -\nu _\varepsilon \quad \text {if} \ {\widetilde{\eta }}\in \mathcal{A}_{{\rho _{*\!*}}+{\rho _{*}}\over 2}^{(\varepsilon )} \, \text {and}\, J_\varepsilon ({\widetilde{\eta }})\in [b-\delta , b+\delta ], \end{aligned}$$

(6.18)

$$\begin{aligned} {d\over dt}J_\varepsilon ({\widetilde{\eta }})&\le -\nu _0 \quad \text {if} \ {\widetilde{\eta }}\in \mathcal{A}_{{{\rho _{*\!*}}+{\rho _{*}}\over 2}}^{(\varepsilon )}\setminus \overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}\, \text {and}\, J_\varepsilon ({\widetilde{\eta }})\in [b-\delta , b+\delta ], \end{aligned}$$

(6.19)

$$\begin{aligned} {d\over dt}T_\varepsilon ({\widetilde{\eta }})&\le 0 \quad \text {if}\ T_\varepsilon ({\widetilde{\eta }}) \ge \kappa _\varepsilon . \end{aligned}$$

(6.20)

The properties (i)–(iii) and (v) follow from the definition (6.16) and the properties (6.17) and (6.20). To complete the proof, we need to show (iv).

We suppose $(z,u)\in \mathcal{A}_{{\rho _{*}}}^{(\varepsilon )}\cap [[J_\varepsilon \le b+\delta ]]$ and we show for some ${\overline{t}}_\varepsilon >0$

$$\begin{aligned} {\widetilde{\eta }}({\overline{t}}_\varepsilon ,z,u)\in [[J_\varepsilon \le b-\delta ]]. \end{aligned}$$

(6.21)

Arguing indirectly, we assume that ${\widetilde{\eta }}(t)\in [[J_\varepsilon >b-\delta ]]$ for all $t\ge 0$. If ${\widetilde{\eta }}(t)={\widetilde{\eta }}(t,z,u)$ satisfies

$$\begin{aligned} {\widetilde{\eta }}(t_0) \in \partial \mathcal{A}_{{\rho _{*\!*}}+{\rho _{*}}\over 2}^{(\varepsilon )} \quad \text {for some}\ t_0>0, \end{aligned}$$

(6.22)

then we can find an interval $[s_{z,u},t_{z,u}]$ such that

$$\begin{aligned}&{\widetilde{\eta }}(t)\in \mathcal{A}_{{\rho _{*\!*}}+{\rho _{*}}\over 2}^{(\varepsilon )}\setminus \mathcal{A}_{{\rho _{*}}}^{(\varepsilon )}\quad \text {for}\ t\in (s_{z,u}, t_{z,u}), \end{aligned}$$

(6.23)

$$\begin{aligned}&{\widetilde{\eta }}(s_{z,u}) \in \partial \mathcal{A}_{{\rho _{*}}}^{(\varepsilon )}, \qquad {\widetilde{\eta }}(t_{z,u})\in \partial \mathcal{A}_{{\rho _{*\!*}}+{\rho _{*}}\over 2}^{(\varepsilon )}. \end{aligned}$$

(6.24)

The following Lemma 6.9 shows that for some $\tau _0>0$ independent of $\varepsilon$, (z, u)

$$\begin{aligned} t_{z,u} - s_{z,u} \ge \tau _0. \end{aligned}$$

(6.25)

Thus by (6.19),

$$\begin{aligned} J_\varepsilon ({\widetilde{\eta }}(t_{z,u})) \le J_\varepsilon ({\widetilde{\eta }}(s_{z,u}))-\nu _0 \tau _0 \le b+\delta -\nu _0\tau _0. \end{aligned}$$

Choosing $\delta <{1\over 3}\nu _0\tau _0$, we have

$$\begin{aligned} J_\varepsilon ({\widetilde{\eta }}(t_{z,u})) \le b-2\delta , \end{aligned}$$

(6.26)

which is in contradiction. Thus (6.22) cannot occur and we have ${\widetilde{\eta }}(t)\in \mathcal{A}_{{\rho _{*\!*}}+{\rho _{*}}\over 2}^{(\varepsilon )}$ for all $t\ge 0$. By (6.18), setting ${\overline{t}}_\varepsilon ={2\delta \over \nu _\varepsilon }>0$, we have (6.21) and (iv) holds. $\square$

The following lemma is a key of the proof of Proposition 6.8. We remark that

$$\begin{aligned} \mathop {\textrm{dist}}\nolimits _{{{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}}(\mathcal{A}_{{\rho _{*}}}^{(\varepsilon )}, \partial \mathcal{A}_{{\rho _{*\!*}}+{\rho _{*}}\over 2}^{(\varepsilon )}) \ge \mathop {\textrm{DIST}}\nolimits _\varepsilon (\mathcal{A}_{{\rho _{*}}}^{(\varepsilon )}, \partial \mathcal{A}_{{\rho _{*\!*}}+{\rho _{*}}\over 2}^{(\varepsilon )}) \ge {1 \over 2}({\rho _{*\!*}}-{\rho _{*}}). \end{aligned}$$

However, since $\ell _\varepsilon \rightarrow \infty$ as $\varepsilon \rightarrow 0$, $\Vert {{d{\widetilde{\eta }}\over dt}}\Vert _{{{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}} =\Vert {W({\widetilde{\eta }})}\Vert _{{{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}}$ is not uniformly bounded by (6.15). Thus (6.25) does not follow from (6.23)–(6.24). In the following lemma, (iv) of Proposition 6.7 plays a role.

Lemma 6.9

There exists $\tau _0>0$ independent of $\varepsilon >0$ such that if ${\widetilde{\eta }}(t)={\widetilde{\eta }}(t,z,u)$ satisfies (6.23)–(6.24), then (6.25) holds.

Proof

By Proposition 6.7 (iv), we have for ${\widetilde{\eta }}(t)={\widetilde{\eta }}(t,z,u)$

$$\begin{aligned} \begin{aligned}&\Vert {{d\over dt}M_1({\widetilde{\eta }}(t))}\Vert _{H^1} \le \varphi _1(J_\varepsilon ({\widetilde{\eta }}))\varphi _2({\widetilde{\eta }})\Vert {DM_1({\widetilde{\eta }})W({\widetilde{\eta }})}\Vert _{H^1} \le C, \\&{d\over dt}\left( \Vert {M_2({\widetilde{\eta }}(t))}\Vert _{H^1}^2\right) = -\varphi _1(J_\varepsilon ({\widetilde{\eta }}))\varphi _2({\widetilde{\eta }})D(\Vert {M_2({\widetilde{\eta }})}\Vert _{H^1}^2)W({\widetilde{\eta }}) \le C'. \end{aligned} \end{aligned}$$

Thus, for $t\in [s_{z,u},s_{z,u}+\tau ]$ we have

$$\begin{aligned}&\Vert {P_2{\widetilde{\eta }}(t)-P_2{\widetilde{\eta }}(s_{z,u})}\Vert _{H^1}\nonumber \\&\quad \le \Vert {M_1(P_2{\widetilde{\eta }}(t))-M_1(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1} + \Vert {M_2(P_2{\widetilde{\eta }}(t))-M_2(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1}\nonumber \\&\quad \le C(t-s_{z,u}) + \Vert {M_2(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1} + \Vert {M_2(P_2{\widetilde{\eta }}(t))}\Vert _{H^1}\nonumber \\&\quad \le C(t-s_{z,u}) + \Vert {M_2(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1} + \left( \Vert {M_2(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1}^2+C'(t-s_{z,u})\right) ^{1/2}\nonumber \\&\quad \le C\tau + \Vert {M_2(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1} + \left( \Vert {M_2(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1}^2+C'\tau \right) ^{1/2}. \end{aligned}$$

(6.27)

On the other hand we have

$$\begin{aligned} \Vert {M_2(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1} \le 3{\rho _{*}}+ d_\varepsilon , \end{aligned}$$

(6.28)

where

$$\begin{aligned} d_\varepsilon = \sup _{\omega \in {\widehat{S}}_b, |{y}|\le R_0} \Vert {(1-\zeta _{1/\sqrt{\varepsilon }}(x))\omega (x-y)}\Vert _{H^1} \rightarrow 0 \quad \text {as}\ \varepsilon \rightarrow 0. \end{aligned}$$

In fact, writing ${\widetilde{\eta }}(s_{z,u})=(z',u')\in \mathcal{A}_{{\rho _{*}}}^{(\varepsilon )}$, we have for some $(\xi _0,\omega _0)\in \overline{\Omega }\times {\widehat{S}}_b$

$$\begin{aligned} \mathop {\textrm{dist}}\nolimits _\varepsilon \left( u'\left( x-{z'\over \varepsilon }\right) , \omega _0\left( x-{\xi _0\over \varepsilon }\right) \right) <{\rho _{*}}. \end{aligned}$$

Thus, there exists $h\in {\mathbb {R}}^N$ such that

$$\begin{aligned} |{h}|^2 + {\left\| u'\left( x-{z'\over \varepsilon }\right) -\omega _0\left( x-{\xi _0+h\over \varepsilon }\right) \right\| }_{H^1}^2 < {\rho _{*}}^2. \end{aligned}$$

By Proposition 3.3, we have $|{\beta (u') - {\xi _0-z'+h\over \varepsilon }}| \le R_0$. Since $P_2{\widetilde{\eta }}(s_{z,u})=u'$, we have by (2.2)

$$\begin{aligned} \begin{aligned}&\Vert {M_2(P_2{\widetilde{\eta }}(s_{z,u}))}\Vert _{H^1} = \Vert {M_2(u')}\Vert _{H^1} =\Vert {(1-\zeta _{1/\sqrt{\varepsilon }}(x-\beta (u')) u'(x)}\Vert _{H^1} \\&\quad \le {\left\| (1-\zeta _{1/\sqrt{\varepsilon }}(x-\beta (u')))\left( u'(x)-\omega _0\left( x-{\xi _0-z+h\over \varepsilon }\right) \right) \right\| }_{H^1} \\&\qquad + {\left\| (1-\zeta _{1\over \sqrt{\varepsilon }}(x-\beta (u')))\omega _0\left( x-{\xi _0-z'+h\over \varepsilon }\right) \right\| }_{H^1} \\&\quad \le 3{\left\| u'(x)-\omega _0\left( x-{\xi _0-z'+h\over \varepsilon }\right) \right\| }_{H^1} +d_\varepsilon \le 3{\rho _{*}}+d_\varepsilon . \end{aligned} \end{aligned}$$

Thus we have (6.28). By (6.27),

$$\begin{aligned} \Vert {P_2{\widetilde{\eta }}(t)-P_2{\widetilde{\eta }}(s_{z,u})}\Vert _{H^1}\le & {} C\tau + (3{\rho _{*}}+d_\varepsilon ) + ((3{\rho _{*}}+d_\varepsilon )^2+C'\tau )^{1/2} \\{} & {} \quad \text {for} \ t\in [s_{z,u},s_{z,u}+\tau ]. \end{aligned}$$

Since $|{P_1W(z,u)}| \le 1$ for all (z, u), we have $|{P_1{\widetilde{\eta }}(t)-P_1{\widetilde{\eta }}(s_{z,u})}| \le \tau$. Thus there exists $\tau _0>0$ such that for $\varepsilon >0$ small

$$\begin{aligned} \begin{aligned} \mathop {\textrm{DIST}}\nolimits _\varepsilon ({\widetilde{\eta }}(t),{\widetilde{\eta }}(s_{z,u})) \le&\Vert {{\widetilde{\eta }}(t)-{\widetilde{\eta }}(s_{z,u})}\Vert _{{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}\\ \le&\left( |{P_1{\widetilde{\eta }}(t)-P_1{\widetilde{\eta }}(s_{z,u})}|^2 +\Vert {P_2{\widetilde{\eta }}(t)-P_2{\widetilde{\eta }}(s_{z,u})}\Vert _{H^1}^2\right) ^{1/2} \\ <&7{\rho _{*}}\quad \text {for}\ t\in [s_{z,u}, s_{z,u}+\tau _0], \end{aligned} \end{aligned}$$

which implies

$$\begin{aligned} \begin{aligned} \mathop {\textrm{DIST}}\nolimits _\varepsilon ({\widetilde{\eta }}(t), \widehat{\mathcal{Z}}_b)&\le \mathop {\textrm{DIST}}\nolimits _\varepsilon ({\widetilde{\eta }}(t),{\widetilde{\eta }}(s_{z,u})) + \mathop {\textrm{DIST}}\nolimits _\varepsilon ({\widetilde{\eta }}(s_{z,u}),\widehat{\mathcal{Z}}_b) \\&<7{\rho _{*}}+{\rho _{*}}< {{\rho _{*\!*}}+{\rho _{*}}\over 2} \quad \text {for}\ t\in [s_{z,u},s_{z,u}+\tau _0]. \end{aligned} \end{aligned}$$

Here we used (4.20). Thus we have ${\widetilde{\eta }}(t)\in \mathcal{A}_{{\rho _{*}}+{\rho _{*\!*}}\over 2}^{(\varepsilon )}$ for $t\in [s_{z,u},s_{z,u}+\tau _0]$ and the proof of Lemma 6.9 is completed. $\square$

End of the proof of Proposition 6.1

We define $\pi _\varepsilon :\, {{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}\rightarrow H^1({\mathbb {R}}^N)$ by

$$\begin{aligned} \pi _\varepsilon (z,u)(x) = u\left( x-{z\over \varepsilon }\right) . \end{aligned}$$

For the flow ${\widetilde{\eta }}(t,z,u)$ obtained in Proposition 6.8, set

$$\begin{aligned} \eta (t,u)=\pi _\varepsilon ({\widetilde{\eta }}(t,0,u)). \end{aligned}$$

Noting $T_\varepsilon (\pi _\varepsilon (z,u))=T_\varepsilon (u)$, it is easily observed that $\eta (t,u)$ has the desired properties. $\square$

7 Existence of Critical Points

In this section we complete a proof of Theorem 1.2. We argue 2 setting (MP) and (LM) separately.

7.1 Existence Under the Condition (MP)

First we consider (1.1) under the assumptions (f1)–(f4), (V1)–(V4) and (MP). Let $V_0>0$ be the number given in (MP) and let $b=E_{V_0}$.

Proposition 7.1

Assume (f1)–(f4), (V1)–(V4) and (MP) and let $b=E_{V_0}$. For any ${\rho _{*}}>0$ and $\overline{\delta }>0$ there exists $\varepsilon _0=\varepsilon _0({\rho _{*}},\overline{\delta })>0$ such that for $\varepsilon \in (0,\varepsilon _0]$, $I_\varepsilon (u)$ has a critical point u in $\overline{N_{{\rho _{*}}}^{(\varepsilon )}}\cap [T_\varepsilon \le \kappa _\varepsilon ]$ with $I_\varepsilon (u)\in [b-\overline{\delta },b+\overline{\delta }]$.

Proof of Proposition 7.1

Let $e_1$, $e_2$, $\Lambda$ be given in (MP). We may choose ${\rho _{*}}>0$ smaller if necessary and choose ${\rho _{*\!*}}>0$ so that (4.20) holds.

Let $\omega _0(x)$ be a least energy solution of $L_{V_0}'(u)=0$. We choose $s_0\in (0,{1 \over 2})$ such that

$$\begin{aligned}&\left\| {\omega _0\left( {x\over s}\right) -\omega _0(x)}\right\| _{H^1} < {{\rho _{*}}\over 3} \quad \text {for all}\ s\in [1-s_0,1+s_0], \end{aligned}$$

(7.1)

$$\begin{aligned}&L_{V(e_i)}\left( \omega _0\left( {x\over s}\right) \right) < b \quad \text {for all}\ s\in [1-s_0,1+s_0] \ \text {and}\ i=0,1. \end{aligned}$$

(7.2)

Since $L_{V(e_i)}(\omega _0({x\over s})) < L_{V_0}(\omega _0({x\over s}))\le b$, (7.2) holds for small $s_0\in (0,{1 \over 2})$.

We may assume that $\overline{\delta }>0$ satisfies

$$\begin{aligned}&\max _{s\in [1-s_0,1+s_0]} L_{V(e_i)}\left( \omega _0\left( {x\over s}\right) \right) < b-2\overline{\delta }\quad \text {for}\ s\in [1-s_0,1+s_0]\ \text {and}\ i=0,1, \end{aligned}$$

(7.3)

$$\begin{aligned}&L_{V_0}\left( \omega _0\left( {x\over s}\right) \right) < b-2\overline{\delta }\quad \text {for}\ s=1\pm s_0. \end{aligned}$$

(7.4)

Arguing indirectly and noting Corollary 5.2, we assume that (6.1) holds. Applying Proposition 6.1, there are $\delta \in (0,\overline{\delta })$ and $\eta (t,u)\in C([0,1]\times A_{{\rho _{*\!*}}}^{(\varepsilon )}, A_{{\rho _{*\!*}}}^{(\varepsilon )})$ such that (i)–(v) of Proposition 6.1 hold.

Step 1: Choice of an initial path $\gamma _{\varepsilon }(s,\xi ):\,{[1-s_0,1+s_0]\times [0,1]}\rightarrow H^1({\mathbb {R}}^N)$

For $c(\xi )\in \Lambda$, we set

$$\begin{aligned} \gamma _{0\varepsilon }(c;s,\xi )(x) = \omega _0\left( {x-c(\xi )/\varepsilon \over s}\right) :\, {[1-s_0,1+s_0]\times [0,1]}\rightarrow H^1({\mathbb {R}}^N). \end{aligned}$$

By the choice (5.15) of $\kappa _\varepsilon$, we have

$$\begin{aligned}&\gamma _{0\varepsilon }(c;s,\xi )\in [T_\varepsilon \le \kappa _\varepsilon ], \end{aligned}$$

(7.5)

$$\begin{aligned}&\gamma _{0\varepsilon }(c;s,\xi )\in A_{{\rho _{*}}}^{(\varepsilon )}\quad \text {for all} \ (s,\xi )\in {[1-s_0,1+s_0]\times [0,1]}. \end{aligned}$$

(7.6)

In fact, $\omega _0(x-c(\xi )/\varepsilon )\in {\widehat{Z}}_b^{(\varepsilon )}$ and (7.1) imply (7.6).

We also have

$$\begin{aligned}{} & {} I_\varepsilon (\gamma _{0\varepsilon }(c;s,\xi )) \rightarrow L\left( c(\xi ),\omega _0\left( {x\over s}\right) \right) \nonumber \\{} & {} \quad = L_{V_0}\left( \omega _0\left( {x\over s}\right) \right) +{1 \over 2}(V(c(\xi ))-V_0)\left\| {\omega _0\left( {x\over s}\right) }\right\| _2^2 \end{aligned}$$

(7.7)

as $\varepsilon \rightarrow 0$ uniformly in ${[1-s_0,1+s_0]\times [0,1]}$.

Thus, choosing $c(\xi )\in \Lambda$ such that $\max _{\xi \in [0,1]} V(c(\xi ))$ is very close to $V_0$, from (7.3), (7.4) and (7.7) we have for sufficiently small $\varepsilon >0$

$$\begin{aligned}&\gamma _{0\varepsilon }(c;s,\xi )\in [I_\varepsilon \le b-\overline{\delta }] \quad \text {for} \ (s,\xi )\in \partial ({[1-s_0,1+s_0]\times [0,1]}), \end{aligned}$$

(7.8)

$$\begin{aligned}&\gamma _{0\varepsilon }(c;s,\xi )\in [I_\varepsilon \le b+\delta ] \quad \text {for}\ (s,\xi )\in {[1-s_0,1+s_0]\times [0,1]}. \end{aligned}$$

(7.9)

Let $\eta (t,u):\, [0,1]\times A_{{\rho _{*\!*}}}^{(\varepsilon )}\rightarrow A_{{\rho _{*\!*}}}^{(\varepsilon )}$ be a deformation given in Proposition 6.1 and we set

$$\begin{aligned} \gamma _{\varepsilon }(s,\xi )=\eta (1,\gamma _{0\varepsilon }(c;s,\xi )). \end{aligned}$$

(7.10)

By (7.8) and the property (ii) of Proposition 6.1,

$$\begin{aligned} \gamma _{\varepsilon }(s,\xi )= & {} \gamma _{0\varepsilon }(c;s,\xi ) =\omega _0\left( {x-c(\xi )/\varepsilon \over s}\right) \nonumber \\{} & {} \text {for}\ (s,\xi )\in \partial ({[1-s_0,1+s_0]\times [0,1]}). \end{aligned}$$

(7.11)

By (7.9) and the properties (iv), (v) of Proposition 6.1, we have for $(s,\xi )\in {[1-s_0,1+s_0]\times [0,1]}$

$$\begin{aligned} \gamma _{\varepsilon }(s,\xi ) \in [I_\varepsilon \le b-\delta ]\cap [T_\varepsilon \le \kappa _\varepsilon ]. \end{aligned}$$

(7.12)

Next we will show under (7.5)–(7.6) and (7.11) that $\gamma _{\varepsilon }(s,\xi )$ satisfies

$$\begin{aligned} \liminf _{\varepsilon \rightarrow 0}\max _{(s,\xi )\in {[1-s_0,1+s_0]\times [0,1]}} I_\varepsilon (\gamma _{\varepsilon }(s,\xi )) \ge b. \end{aligned}$$

(7.13)

We note that (7.13) is incompatible with (7.12) and it shows the existence of a critical point in $\overline{N_{{\rho _{*}}}^{(\varepsilon )}}\cap [T_\varepsilon \le \kappa _\varepsilon ]$.

We remark that under (MP) there exists a small neighborhood $\Omega '(\supset \Omega )$ of $\Omega$ with the following properties:

• (1) For $\varepsilon >0$ small,

  $$\begin{aligned} \varepsilon \beta (\gamma _{\varepsilon }(s,\xi ))\in \Omega ' \quad \text {for all}\ (s,\xi )\in [1-s_0,1+s_0]\times [0,1]. \end{aligned}$$

• (2) Set

  $$\begin{aligned} W=\{x \in \Omega ':\, V(x)<V_0\}, \end{aligned}$$

  then $e_0$ and $e_1$ belong to different components of W.

Since $\gamma _{\varepsilon }(s,z)\in \mathcal{A}_{{\rho _{*\!*}}}^{(\varepsilon )}$ for all (s, z), we have

$$\begin{aligned} \mathop {\textrm{dist}}\nolimits _{{\mathbb {R}}^N}(\varepsilon \beta (\gamma _{\varepsilon }(s,\xi )),{\overline{\Omega }}) \le \varepsilon R_0+{\rho _{*\!*}}\end{aligned}$$

and (1) follows.

We denote by $W_-$ the component of W, to which $e_0$ belongs, and we set

$$\begin{aligned} W_+= W\setminus W_-, \qquad W_0=\{ x\in \Omega ':\, V(x)\ge V_0\}. \end{aligned}$$

We also introduce a signed distance function $d_0(x)$ on $\Omega '$ by

$$\begin{aligned} d_0(x)={\left\{ \begin{array}{ll} -{\mathop {\textrm{dist}}\nolimits (x, W_0)\over \mathop {\textrm{dist}}\nolimits (e_0, W_0)} &{}\text {if}\ x\in W_-,\\ {\mathop {\textrm{dist}}\nolimits (x, W_0)\over \mathop {\textrm{dist}}\nolimits (e_1, W_0)} &{}\text {if}\ x\in W_+,\\ 0 &{}\text {if}\ x\in W_0. \end{array}\right. } \end{aligned}$$

For $P_a(u)$ defined in (3.2), we set $a=V_0$ and consider

$$\begin{aligned} F_\varepsilon (u)=(P_{V_0}(u), d_0(\varepsilon \beta (u))):\, A_{{\rho _{*\!*}}}^{(\varepsilon )}\rightarrow {\mathbb {R}}\times {\mathbb {R}}. \end{aligned}$$

Then we have

Step 2: For $\gamma _{\varepsilon }(s,\xi )$ defined in (7.10),

$$\begin{aligned} \mathop {\textrm{deg}}\nolimits (F_\varepsilon (\gamma _{\varepsilon }(s,\xi )),[1-s_0,1+s_0]\times [0,1],(0,0))=-1. \end{aligned}$$

(7.14)

In particular, there exists $(s_\varepsilon ,\xi _\varepsilon )\in [1-s_0,1+s_0]\times [0,1]$ such that

$$\begin{aligned} P_{V_0}(\gamma _{\varepsilon }(s_\varepsilon ,\xi _\varepsilon ))=0 \quad \text {and} \quad V(\varepsilon \beta (\gamma _{\varepsilon }(s_\varepsilon ,\xi _\varepsilon ))) \ge V_0. \end{aligned}$$

(7.15)

In fact, for $(s,\xi )\in \partial ([1-s_0,1+s_0]\times [0,1])$, we have by (7.11)

$$\begin{aligned} \begin{aligned} F_\varepsilon (\gamma _{\varepsilon }(s,\xi ))&= F_\varepsilon \left( \omega _0\left( {x-c(\xi )/\varepsilon \over s}\right) \right) \\&= \left( P_{V_0}\left( \omega _0\left( {x\over s}\right) \right) , d_0\left( \varepsilon \beta \left( \omega _0\left( {x-c(\xi )/\varepsilon \over s}\right) \right) \right) \right) \\&= \left( P_{V_0}\left( \omega _0\left( {x\over s}\right) \right) , d_0(c(\xi )+o(1))\right) . \end{aligned} \end{aligned}$$

By Proposition 3.2 we have

$$\begin{aligned} P_{V_0}\left( \omega _0\left( {x\over s}\right) \right) {\left\{ \begin{array}{ll}>0 &{}\text {for}\ s=1-s_0,\\<0 &{}\text {for}\ s=1+s_0, \end{array}\right. } \qquad d_0(c(\xi )) {\left\{ \begin{array}{ll} >0 &{}\text {for}\ \xi =0,\\ <0 &{}\text {for}\ \xi =1, \end{array}\right. } \end{aligned}$$

and thus we have (7.14). Since $d_0(y)=0$ implies $V(y)\ge V_0$, (7.14) implies the existence of $(s_\varepsilon ,\xi _\varepsilon )$ with (7.15).

Step 3: $I_\varepsilon (\gamma _{\varepsilon }(s_\varepsilon ,\xi _\varepsilon ))\ge b+o(1)$ as $\varepsilon \rightarrow 0$.

We write $w_\varepsilon =\gamma _{\varepsilon }(s_\varepsilon ,\xi _\varepsilon )$. Since $w_\varepsilon \in A_{{\rho _{*\!*}}}^{(\varepsilon )}\cap [T_\varepsilon \le \kappa _\varepsilon ]$, it follows from Proposition 5.4

$$\begin{aligned} \begin{aligned} I_\varepsilon (w_\varepsilon )&\ge L(V(\varepsilon \beta (w_\varepsilon )),w_\varepsilon ) -c_\varepsilon -{1 \over 2}{\overline{V}}\kappa _\varepsilon \\&\ge L_{V_0}(w_\varepsilon )+{1 \over 2}(V(\varepsilon \beta (w_\varepsilon ))-V_0)\Vert {w_\varepsilon }\Vert _2^2 -c_\varepsilon -{1 \over 2}{\overline{V}}\kappa _\varepsilon . \end{aligned} \end{aligned}$$

By (7.15), we have

$$\begin{aligned} I_\varepsilon (w_\varepsilon ) \ge L_{V_0}(w_\varepsilon )-c_\varepsilon -{1 \over 2}{\overline{V}}\kappa _\varepsilon . \end{aligned}$$

By (3.3), it follows from $P_{V_0}(w_\varepsilon )=0$ that $L_{V_0}(w_\varepsilon ) \ge E_{V_0}=b$. Thus we have Step 3.

Step 4: Conclusion.

(7.12) and (7.13) are incompatible and thus (6.1) does not hold. Thus we have the conclusion of Proposition 7.1. $\square$

7.2 Existence Under the Condition (LM)

In this section we consider (1.1) under the assumptions (f1)–(f4), (V1)–(V4) and (LM). Let $V_0>0$ be the maximum in $\Omega$ and let $b=E_{V_0}$. We have

Proposition 7.2

Assume (f1)–(f4), (V1)–(V4) and (LM) and let $b=E_{V_0}$. For any ${\rho _{*}}>0$ and $\overline{\delta }>0$ there exists $\varepsilon _0=\varepsilon _0({\rho _{*}},\overline{\delta })>0$ such that for $\varepsilon \in (0,\varepsilon _0]$, $I_\varepsilon (u)$ has a critical point u in $\overline{N_{{\rho _{*}}}^{(\varepsilon )}}\cap [T_\varepsilon \le \kappa _\varepsilon ]$ satisfying $I_\varepsilon (u)\in [b-\overline{\delta },b+\overline{\delta }]$.

Proof of Proposition 7.2

Let $\omega _0(x)$ be a least energy solution corresponding to $b=E_{V_0}$. We choose $s_0\in (0,{1 \over 2})$ satisfying (7.1) and set $\gamma _{0\varepsilon }(s,\xi ):\, [1-s_0,1+s_0]\times {\overline{\Omega }}\rightarrow H^1({\mathbb {R}}^N)$ by

$$\begin{aligned} \gamma _{0\varepsilon }(s,\xi )(x)=\omega _0\left( {x-\xi /\varepsilon \over s}\right) . \end{aligned}$$

We note that

$$\begin{aligned} \begin{aligned} I_\varepsilon (\gamma _{0\varepsilon }(s,\xi ))&= {1 \over 2}\left\| {\nabla \left( \omega _0\left( {x\over s}\right) \right) }\right\| _2^2 +{1 \over 2}\int _{{\mathbb {R}}^N}V(\varepsilon x+\xi ) \omega _0\left( {x\over s}\right) ^2 -{1 \over 2}\mathcal{D}\left( \omega _0\left( {x\over s}\right) \right) \\&\rightarrow L\left( \xi ,\omega _0\left( {x\over s}\right) \right) = L_{V_0}\left( \omega _0\left( {x\over s}\right) \right) -{1 \over 2}(V_0-V(\xi ))\left\| {\omega _0\left( {x\over s}\right) }\right\| _2^2 \end{aligned} \end{aligned}$$

as $\varepsilon \rightarrow 0$ uniformly in $(s,\xi )\in {[1-s_0,1+s_0]\times {\overline{\Omega }}}$.

Thus there exists $\overline{\delta }>0$ such that

$$\begin{aligned} \begin{aligned}&\max _{(s,\xi )\in {[1-s_0,1+s_0]\times {\overline{\Omega }}}} L\left( \xi ,\omega _0\left( {x\over s}\right) \right) \le b, \\&\max _{(s,\xi )\in \partial ({[1-s_0,1+s_0]\times {\overline{\Omega }}})} L\left( \xi ,\omega _0\left( {x\over s}\right) \right) \le b-2\overline{\delta }. \end{aligned} \end{aligned}$$

Moreover for any $\delta \in (0,\overline{\delta })$ we have for sufficiently small $\varepsilon >0$

$$\begin{aligned} \begin{aligned}&\max _{(s,\xi )\in {[1-s_0,1+s_0]\times {\overline{\Omega }}}} I_\varepsilon (\gamma _{0\varepsilon }(s,\xi )) \le b+\delta , \\&\max _{(s,\xi )\in \partial ({[1-s_0,1+s_0]\times {\overline{\Omega }}})} I_\varepsilon (\gamma _{0\varepsilon }(s,\xi )) \le b-\overline{\delta }. \end{aligned} \end{aligned}$$

We also note that $\gamma _{0\varepsilon }(s,\xi )\in [T_\varepsilon \le \kappa _\varepsilon ]$ for all $(s,\xi )\in {[1-s_0,1+s_0]\times {\overline{\Omega }}}$. We define $F_\varepsilon :A_{{\rho _{*\!*}}}^{(\varepsilon )}\rightarrow {\mathbb {R}}\times {\mathbb {R}}^N$ by

$$\begin{aligned} F_\varepsilon (u)=(P_{V_0}(u), \varepsilon \beta (u)). \end{aligned}$$

Arguing as in the proof of Proposition 7.1, we can prove Proposition 7.2. $\square$

7.3 End of the Proof of Theorem 1.2

Finally we derive our Theorem 1.2 from Propositions 7.1 and 7.2.

End of the proof of Theorem 1.2

Let $V_0$ be the critical value given by (MP) or (LM). Since $V(x)\in C^N({\mathbb {R}}^N,{\mathbb {R}})$, by the Sard Theorem there exists a sequence $(\alpha _n)_{n=1}^\infty \subset (0,\infty )$ such that

• (1) $\alpha _1>\alpha _2>\cdots> \alpha _n>\alpha _{n+1}>\cdots$;

• (2) $\alpha _n\rightarrow 0$ as $n\rightarrow \infty$;

• (3) $V_0-\alpha _n$ is a regular value of V(x).

We set

$$\begin{aligned} \Omega _n=\{x\in \Omega :\, V(x)>V_0-\alpha _n\}. \end{aligned}$$

We can see that (V1)–(V4) and (MP) or (LM) hold in $\Omega _n$ for large n (See Sect. 4.1). Thus we can apply the arguments in previous sections in $\Omega _n$ and, replacing $\Omega$ with $\Omega _n$, we prove Propositions 7.1 or 7.2 for $\Omega _n$. That is, for any ${\rho _{*}}>0$ and $\overline{\delta }>0$ there exists $\varepsilon _0(n,{\rho _{*}},\overline{\delta })>0$ such that for $\varepsilon \in (0,\varepsilon _0(n,{\rho _{*}},\overline{\delta })]$, $I_\varepsilon (u)$ has a critical point $u_\varepsilon$ in $N_{n,\rho _*}^{(\varepsilon )}$ with $I_\varepsilon (u_\varepsilon )\in [b-\overline{\delta },b+\overline{\delta }]$. Precisely,

$$\begin{aligned} \begin{aligned} {\widehat{\mathcal{K}}}_{b,n}&=\{ (\xi ,\omega ):\, \xi \in \Omega _n,\, DL(\xi ,\omega )=0,\, L(\xi ,\omega )=b,\, \Vert {\omega }\Vert _{L^2(Q)}=\max _{n\in {\mathbb {Z}}^N}\Vert {\omega }\Vert _{L^2(n+Q)}\} \\&=\{ (\xi ,\omega ):\, \xi \in \Omega ,\, V(\xi )\in [V_0-\alpha _n,V_0],\, \nabla V(\xi )=0,\, L_{V(\xi )}'(\omega )=0,\\&\qquad L_{V(\xi )}(\omega )=b,\, \Vert {\omega }\Vert _{L^2(Q)}=\max _{n\in {\mathbb {Z}}^N}\Vert {\omega }\Vert _{L^2(n+Q)}\}, \\ {\widehat{K}}_{b,n}^{(\varepsilon )}&= \left\{ \omega \left( x-{\xi \over \varepsilon }\right) :\, (\xi ,\omega )\in {\widehat{\mathcal{K}}}_{b,n}\right\} , \\ N_{n,{\rho _{*}}}^{(\varepsilon )}&= \{ u\in H^1({\mathbb {R}}^N):\, \mathop {\textrm{dist}}\nolimits _\varepsilon (u, \widehat{K}_{b,n}^{(\varepsilon )})<{\rho _{*}}\}. \end{aligned} \end{aligned}$$

We note that ${\widehat{\mathcal{K}}}_{b,n}$ shrinks to the following ${\widehat{\mathcal{K}}}_{b,\infty }$ as $n\rightarrow \infty$:

$$\begin{aligned} \begin{aligned} {\widehat{\mathcal{K}}}_{b,\infty }&=\{ (\xi ,\omega ):\, \xi \in \Omega ,\, V(\xi )=V_0,\, DL(\xi ,\omega )=0,\, L(\xi ,\omega )=b,\\&\qquad \Vert {\omega }\Vert _{L^2(Q)}=\max _{n\in {\mathbb {Z}}^N}\Vert {\omega }\Vert _{L^2(n+Q)}\} \\&=\textrm{Crit}_{V_0} \times \{\omega \in H^1({\mathbb {R}}^N):\, L_{V_0}'(\omega )=0,\, L_{V_0}(\omega )=b,\\&\qquad \Vert {\omega }\Vert _{L^2(Q)} =\max _{n\in {\mathbb {Z}}^N}\Vert {\omega }\Vert _{L^2(n+Q)}\}. \end{aligned} \end{aligned}$$

That is, $\mathop {\textrm{dist}}\nolimits _{{{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}}({\widehat{\mathcal{K}}}_{b,n}, {\widehat{\mathcal{K}}}_{b,\infty }) \rightarrow 0$ as $n\rightarrow \infty$. Now we can complete the proof of Theorem 1.2. We choose sequences $(\rho _{*n})_{n=1}^\infty$, $(\overline{\delta }_n)_{n=1}^\infty$ with $\rho _{*n}\rightarrow 0$, $\overline{\delta }_n\rightarrow 0$ as $n\rightarrow \infty$. Then there exists $\varepsilon _n=\varepsilon _0(n,\rho _{*n},\overline{\delta }_n)>0$ such that for $\varepsilon \in (0,\varepsilon _n]$, $I_\varepsilon (u)$ has a critical point $u_{n\varepsilon }\in N_{n,\rho _{*n}}^{(\varepsilon )}$ with $I_\varepsilon (u_{n\varepsilon })\in [b-\overline{\delta }_n,b+\overline{\delta }_n]$. We may assume $\varepsilon _1>\varepsilon _2>\cdots>\varepsilon _n>\varepsilon _{n+1}>\cdots$ and $\varepsilon _n\rightarrow 0$ as $n\rightarrow \infty$. Finally we set

$$\begin{aligned} u_\varepsilon (x)=u_{n\varepsilon }(x) \quad \text {for} \ \varepsilon \in (\varepsilon _{n-1},\varepsilon _n]. \end{aligned}$$

We observe that $(u_\varepsilon )_{\varepsilon \in (0,\varepsilon _1]}$ is the desired family of solutions. $\square$

Proof of Theorem 1.3

Under the assumptions (V1’) and (V1”), V(x) has finitely many critical points in $\Omega$. So there exists $\alpha >0$ such that there are no critical values of $V\mid _\Omega$ in $[V_0-\alpha , V_0+\alpha ]{\setminus }\{ V_0\}$. Replacing $\Omega$ with

$$\begin{aligned} \{ x\in \Omega :\, V(x)\in (V_0-\alpha , V_0+\alpha )\} \end{aligned}$$

and arguing as in Sect. 4.1, we may assume that $x\in \Omega$ and $\nabla V(x)=0$ imply $V(x)=V_0$. Thus for $b=E_{V_0}$

$$\begin{aligned} \widehat{\mathcal{K}}_b=\text {Crit}_{V_0}\times \mathcal{C}_b, \end{aligned}$$

where $\mathcal{C}_b$ is a set of least energy solutions of $L_{V_0}(u)=0$, that is,

$$\begin{aligned} \mathcal{C}_b=\{ \omega \in H^1({\mathbb {R}}^N):\, L_{V_0}(\omega )=b, \, L_{V_0}'(\omega )=0, \, \Vert {\omega }\Vert _{L^2(Q)}=\max _{n\in {\mathbb {N}}^N}\Vert {\omega }\Vert _{L^2(n+Q)} \}. \end{aligned}$$

Thus $N_\rho ^{(\varepsilon )}$ is a $\rho$-neighborhood of

$$\begin{aligned} {\widehat{K}}_b^{(\varepsilon )}=\left\{ \omega \left( x-{\xi \over \varepsilon }\right) :\, \xi \in \text {Crit}_{V_0},\, \omega \in \mathcal{C}_b\right\} . \end{aligned}$$

By the arguments in the proof of Propositions 7.1 and 7.2, for any ${\rho _{*}}$ and $\overline{\delta }>0$ there exists $\varepsilon _0=\varepsilon _0({\rho _{*}},\overline{\delta })>0$ such that for $\varepsilon \in (0,\varepsilon _0]$, $I_\varepsilon (u)$ has a critical point in $N_{{\rho _{*}}}^{(\varepsilon )}$.

Taking sequences $(\rho _{*n})_{n=1}^\infty$, $(\overline{\delta }_n)_{n=1}^\infty$ with $\rho _{*n}\rightarrow 0$, $\overline{\delta }_n\rightarrow 0$ and arguing as in the proof of Theorem 1.2, we complete the proof of Theorem 1.3. $\square$

7.4 Potential V(x) of Class $C^1$

In previous sections we consider the situation where the set of critical values $\{ V(x):\, x\in \Omega ,\, \nabla V(x)=0\}$ is of measure 0, which is ensured by Sard Theorem for $V(x)\in C^N({\mathbb {R}}^N,{\mathbb {R}})$. In this section we assume just $V(x)\in C^1({\mathbb {R}}^N,{\mathbb {R}})$. Then the set of critical values may not be of measure 0.

We have the following weaker result.

Theorem 7.3

Assume (f1)–(f4) and (V1’), (V2), (V3). Moreover suppose (LM) or (MP). Moreover assume (V4) in Sect. 4.1 for a constant $V_0$ appeared in (LM) or (MP). Then (1.1) has a family of solutions, which concentrates in $\Omega$. That is, there exists $\varepsilon _0>0$ and a family $(u_\varepsilon )_{\varepsilon \in (0, \varepsilon _0]}$ of solutions of (1.2) with the following property: for any sequence $(\varepsilon _j)_{j=1}^\infty \subset (0,\varepsilon _0]$ with $\varepsilon _j\rightarrow 0$ after extracting a subsequence — still we denote it by $\varepsilon _j$ — there exist $(x_j)_{j=1}^\infty \subset {\mathbb {R}}^N$, $x_0\in \Omega$ and a non-trivial solution $\omega _0(x)\in H^1({\mathbb {R}}^N)$ of the limit problem $-\Delta u+V(x_0)u = (I_\alpha *F(u))F'(u)$ in ${\mathbb {R}}^N$ such that

$$\begin{aligned} \varepsilon _j x_j \rightarrow x_0, \quad u_j(x+x_j)\rightarrow \omega _0(x) \ \text {strongly in}\ H^1({\mathbb {R}}^N)\ \text {as} \ j\rightarrow \infty . \end{aligned}$$

Moreover, $(x_0,\omega _0)$ satisfies for $b=E_{V_0}$

$$\begin{aligned} \nabla V(x_0)=0, \quad V(x_0)\le V_0, \quad \partial L(x_0,\omega _0) =0, \quad L(x_0,\omega _0)=b. \end{aligned}$$

In Theorem 7.3, the concentration point $x_0$ is a critical point of V(x) in $\Omega$ but its critical level may be lower than $V_0$ in general.

Proof of Theorem 7.3

For $V_0>0$ given in (LM) or (MP) and let $b=E_{V_0}>0$ be a least energy level for the limit functional $L_{V_0}(u)$. As in the previous sections, we set

$$\begin{aligned} \begin{aligned} \widehat{\mathcal{K}}_b= \{&(\xi ,\omega )\in \Omega \times H^1({\mathbb {R}}^N):\, DL(\xi ,\omega )=0, \, L(\xi ,\omega )=b, \\&\quad \Vert {\omega }\Vert _{L^2(Q)} = \max _{n\in {\mathbb {Z}}^N}\Vert {\omega }\Vert _{L^2(n+Q)} \}. \end{aligned} \end{aligned}$$

Then, following the proofs of Proposition 7.1 and 7.2, let $0<{\rho _{*}}<{\rho _{*\!*}}$ be the numbers satisfying (4.20). For any $\overline{\delta }>0$ there exists $\varepsilon _0({\rho _{*}},\overline{\delta })>0$ such that for $\varepsilon \in (0,\varepsilon _0]$, $I_\varepsilon (u)$ has a critical point u in $\overline{\mathcal{N}_{{\rho _{*}}}^{(\varepsilon )}}$ satisfying $I_\varepsilon (u)\in [b-\overline{\delta },b+\overline{\delta }]$.

Choosing sequences $(\rho _{*n})_{n=1}^\infty$, $(\overline{\delta }_n)_{n=1}^\infty$ with $\rho _{*n}\rightarrow 0$, $\overline{\delta }_n\rightarrow 0$, we complete the proof of Theorem 7.3. $\square$

8 Concentration at a Local Minimum

In Sects. 1, 2, 3, 4, 5, and 6, we develop a deformation theory under our new version of Palais-Smale condition (see Proposition 4.5), i.e., if $(\varepsilon _j)_{j=1}^\infty \subset (0,1]$ and $u_j\in A_{\rho _3}^{(\varepsilon _j)}$ satisfy as $j\rightarrow \infty$

$$\begin{aligned}&\varepsilon _j\rightarrow 0, \quad I_{\varepsilon _j}(u_j)\rightarrow b, \quad I_{\varepsilon _j}'(u_j)\rightarrow 0 \ \text {in}\ (H^1({\mathbb {R}}^N))^*, \end{aligned}$$

(8.1)

$$\begin{aligned}&H_{\varepsilon _j}(u_j)\rightarrow 0, \end{aligned}$$

(8.2)

then

$$\begin{aligned} \mathop {\textrm{dist}}\nolimits _{\varepsilon _j}(u_j, {\widehat{K}}_b^{(\varepsilon _j)}) \rightarrow 0. \end{aligned}$$

(8.3)

And our deformation flow $\eta (t,u)$ is constructed through a deformation in the augmented space ${{\mathbb {R}}^N\times H^1({\mathbb {R}}^N)}$. When a stronger version of Palais-Smale condition, i.e., if (8.3) holds under (8.1) (without (8.2)), we can construct the desired flow directly as a deformation in $H^1({\mathbb {R}}^N)$.

We note that for the functional $I_\varepsilon (u)$ corresponding to the nonlinear Choquard equation (1.2) under the conditions (f1)–(f4), (V2) and

($\widetilde{\hbox {V1}}$):

$V\in C({\mathbb {R}}^N,{\mathbb {R}})$;

($\widetilde{\hbox {LM}}$):

There exists a bounded connected open set $\Omega \subset {\mathbb {R}}^N$ such that

$$\begin{aligned} V_0 \equiv \inf _{x\in \Omega } V(x) < \inf _{x\in \partial \Omega } V(x), \end{aligned}$$

the compactness (8.3) holds under (8.1). This fact is essentially given in Proposition 4.1 in [23].

In fact, if (8.3) holds under (8.1) and if

$$\begin{aligned} I_\varepsilon '(u)\not =0 \quad \text {for all}\ u\in N_{{\rho _{*}}}^{(\varepsilon )}\ \text {with}\ I_\varepsilon (u)\in [b-\delta _0,b+\delta _0], \end{aligned}$$

then for any $\rho _*$, $\rho _{**}>0$ with (4.20) and for $\varepsilon >0$ small there exist constants $\nu _\varepsilon >0$ depending on $\varepsilon$ and $\nu _0>0$ independent of $\varepsilon$ and a locally Lipschitz continuous vector field

$$\begin{aligned} W(u):\, A_{{\rho _{*\!*}}}^{(\varepsilon )}\rightarrow H^1({\mathbb {R}}^N)\end{aligned}$$

such that

• (i) For $T_\varepsilon (u):\, {\widehat{S}}_{b,\rho _\varepsilon }\rightarrow {\mathbb {R}}$ defined (5.1),

  $$\begin{aligned} T_\varepsilon '(u)W(u)>0 \quad \text {if}\ T_\varepsilon (u)\ge \kappa _\varepsilon . \end{aligned}$$

• (ii) For $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}$ with $I_\varepsilon (u)\in [b-\delta _0,b+\delta _0]$

  $$\begin{aligned} I_\varepsilon '(u)W(u) \ge \nu _\varepsilon . \end{aligned}$$

• (iii) For $u\in A_{{\rho _{*\!*}}}^{(\varepsilon )}\setminus \overline{N_{{\rho _{*}}}^{(\varepsilon )}}$ with $I_\varepsilon (u)\in [b-\delta _0,b+\delta _0]$

  $$\begin{aligned} I_\varepsilon '(u)W(u) \ge \nu _0. \end{aligned}$$

• (iv) There exist C, $C'>0$ such that for $M_1(u)$, $M_2(u)$ given in (5.3)

  $$\begin{aligned} \begin{aligned}&\Vert { M_1'(u)W(u)}\Vert _{H^1} <C, \\&\partial _u(\Vert {M_2(u)}\Vert _{H^1}^2)W(u) >-C'. \end{aligned} \end{aligned}$$

Here we use the arguments in Sects. 5 and 6. We obtain a deformation flow $\eta (t,u):\, [0,1]\times A_{{\rho _{*\!*}}}^{(\varepsilon )}\rightarrow A_{{\rho _{*\!*}}}^{(\varepsilon )}$ with the properties (i)–(v) in Proposition 6.1 as a solution of ODE in $H^1({\mathbb {R}}^N)$:

$$\begin{aligned} {d\eta \over dt}=-\varphi _1(I_\varepsilon (\eta ))\varphi _2(\eta ) W(\eta ), \qquad \eta (0,u)=u, \end{aligned}$$

where $\varphi _1(s):\, {\mathbb {R}}\rightarrow [0,1]$, $\varphi _2(u):\, H^1({\mathbb {R}}^N)\rightarrow [0,1]$ are suitable cut-off functions. Thus we have the following result.

Theorem 8.1

(Theorem 1.1 of [23]). Assume the conditions (f1)–(f4) and ($\widetilde{\hbox {V1}}$), (V2), ($\widetilde{\hbox {LM}}$). Then (1.1) has at least one positive solution concentrating in $\Omega$.

Remark 8.2

In [23], we study the existence of solutions of (1.1) concentrating in a potential well $\Omega$, i.e., under ($\widetilde{\hbox {LM}}$) using 2 flows; one flow is the standard gradient flow corresponding to $-I_\varepsilon '(u)$ and the other is the tail minimizing flow. We can give a simplified proof to the result in [23] using our deformation flow $\eta (t,u)$, which keeps the size $T_\varepsilon (u)$ of tail of functions small and we can show the existence of critical points using just one flow $\eta (t,u)$. We note that in [23] we also study the multiplicity of solutions using cup length of the critical set $K=\{ x\in \Omega :\, V(x)=V_0\}$.

Remark 8.3

Our deformation argument can be applied to various singular perturbation problems. For example, it is applicable to the following nonlinear Schrödinger equations:

$$\begin{aligned} -\varepsilon ^2 \Delta u +V(x)u = g(u) \quad \text {in}\ {\mathbb {R}}^N, \end{aligned}$$

(8.4)

where $N\ge 2$, $g(\xi )\in C({\mathbb {R}},{\mathbb {R}})$.

We can use our new deformation argument to improve results in [8] slightly and to simplify the proofs and arguments (c.f. [26]). In [8], Byeon and the second author studied (8.4) under the assumption $g(\xi )\in C^1({\mathbb {R}},{\mathbb {R}})$, which is used to solve elliptic problems (1.6) outside of a large ball uniquely. By virtue of our new deformation flow obtained in Proposition 6.1, which keeps the $H^1$-energy small outside a ball, we don’t need to solve the elliptic problems outside of a ball uniquely and we can relax the regularity assumption on g to the class $C^0$.