Multiplicity and concentration of solutions to the nonlinear magnetic Schrödinger equation

Abstract

In this paper, we study the following nonlinear magnetic Schrödinger equation

$$\begin{aligned} \left\{ \begin{aligned}&\Big (\frac{\varepsilon }{i}\nabla -A(x)\Big )^{2}u+V(x)u=f(|u|^{2})u \quad \hbox {in }{\mathbb {R}}^N\ (N\ge 2),\\&u\in H^{1}({\mathbb {R}}^{N}, {\mathbb {C}}), \end{aligned} \right. \end{aligned}$$

where $\epsilon$ is a positive parameter, and $V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$, $A: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$ are continuous potentials. Under a local assumption on the potential V, by combining variational methods, penalization techniques, and the Ljusternik–Schnirelmann theory, we prove multiplicity and concentration properties of solutions for $\varepsilon >0$ small. In our problem, the function f is only continuous, which allows to consider larger classes of nonlinearities in the reaction.

1 Introduction and main results

The Schrödinger equation is central in quantum mechanics and it plays the role of Newton’s laws and conservation of energy in classical mechanics, that is, it predicts the future behaviour of a dynamical system. It is striking to point out that talking about his celebrated equation, Erwin Schrödinger said: “I don’t like it, and I’m sorry I ever had anything to do with it”. The linear Schrödinger equation is a central tool of quantum mechanics, which provides a thorough description of a particle in a non-relativistic setting. Schrödinger’s linear equation is

$$\begin{aligned} \nabla ^2\psi +\frac{8\pi ^2m}{\hbar ^2}\,\left( E-V(x)\right) \psi =0\,, \end{aligned}$$

where $\psi$ is the Schrödinger wave function, m is the mass of the particle, $\hbar$ denotes Planck’s renormalized constant, E is the energy, and V stands for the potential energy.

Schrödinger also established the classical derivation of his equation, based upon the analogy between mechanics and optics, and closer to de Broglie’s ideas. He developed a perturbation method, inspired by the work of Lord Rayleigh in acoustics, proved the equivalence between his wave mechanics and Heisenberg’s matrix, and introduced the time dependent Schrödinger’s equation

$$\begin{aligned} i\hbar \psi _t=-\frac{\hbar ^2}{2m}\,\nabla ^2\psi +V(x)\psi -\gamma |\psi |^{p-1}\psi \qquad x\in {{\mathbb {R}}}^N \text{( }N\ge 2\text{) },\end{aligned}$$

(1.1)

where $p<2N/(N-2)$ if $N\ge 3$ and $p<+\infty$ if $N=2$.

In physical problems, a cubic nonlinearity corresponding to $p = 3$ in equation (1.1) is common; in this case, problem (1.1) is called the Gross-Pitaevskii equation. In the study of equation (1.1), Floer, Weinstein [22] and Oh [30, 31] supposed that the potential V is bounded and possesses a non-degenerate critical point at $x=0$. More precisely, it is assumed that V belongs to the class ($V_a$) (for some real number a) introduced by Kato [27]. Taking $\gamma >0$ and $\hbar >0$ sufficiently small and using a Lyapunov-Schmidt type reduction, Oh [30, 31] proved the existence of bound state solutions of problem (1.1), that is, solutions of the form

$$\begin{aligned} \psi (x,t)=e^{-iEt/\hbar }u(x)\,. \end{aligned}$$

(1.2)

Using the Ansatz (1.2), we reduce the nonlinear Schrödinger equation (1.1) to the semilinear elliptic equation

$$\begin{aligned} -\frac{\hbar ^2}{2m}\,\nabla ^2 u+\left( V(x)-E\right) u=|u|^{p-1}u\,. \end{aligned}$$

The change of variable $y=\hbar ^{-1}x$ (and replacing y by x) yields

$$\begin{aligned} -\nabla ^2 u+2m\left( V_\hbar (x)-E\right) u=|u|^{p-1}u\qquad x\in {{\mathbb {R}}}^N\,, \end{aligned}$$

(1.3)

where $V_\hbar (x)=V(\hbar x)$.

1.1 Related results

In this paper, we are concerned with multiplicity and concentration results for the following nonlinear magnetic Schrödinger equation

$$\begin{aligned} \Big (\frac{\varepsilon }{i}\nabla -A(x)\Big )^{2}u+V(x)u=f(|u|^{2})u \quad \hbox {in }{\mathbb {R}}^N\ (N\ge 2), \end{aligned}$$

(1.4)

where $u\in H^{1}({\mathbb {R}}^{N}, {\mathbb {C}})$, $\varepsilon >0$ is a parameter, $V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}$ is a continuous function, $f\in C({\mathbb {R}}, {\mathbb {R}})$, and the magnetic potential $A: {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$ is Hölder continuous with exponent $\alpha \in (0, 1]$.

Problem (1.4) arises when one looks for standing wave solutions $\psi (x, t):=e^{-iEt/\hbar }u(x)$ (with $E\in {\mathbb {R}}$) of the nonlinear evolution system

$$\begin{aligned} i\hbar \frac{\partial \psi }{\partial t}=\left( \frac{\hbar }{i}\nabla -A(x)\right) ^{2}\psi +U(x)\psi -f(|\psi |^{2})\psi \quad \hbox {in }{\mathbb {R}}^N\times {\mathbb {R}}. \end{aligned}$$

From a physical point of view, the existence of such solutions and the study of their shape in the semiclassical limit, namely, as $\hbar \rightarrow 0^{+}$ (or, equivalently, as $\varepsilon \rightarrow 0^+$ in (1.4)), is of the greatest importance, since the transition from quantum mechanics to classical mechanics can be formally performed by sending to zero the Planck constant $\hbar$.

For problem (1.4), there is a vast literature concerning the existence and the multiplicity of bound state solutions for the case without magnetic field, namely if $A\equiv 0$. The first result in this direction was given by Floer and Weinstein [22], who considered the case $N=1$ and $f=i_{\mathbb {R}}$. Later on, several authors generalized this result to larger values of N, using different methods. For instance, del Pino and Felmer [20] studied the existence and concentration of solutions to the following problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^{2}\nabla ^2 u+V(x)u=f(u) \hbox { in }\Omega ,\\ u=0 \text { on } \partial \Omega , \\ u>0 \hbox { in }\Omega , \end{array}\right. } \end{aligned}$$

where $\Omega$ is a possibly unbounded domain in ${\mathbb {R}}^{N}$ ($N\ge 3$), the potential V is locally Hölder continuous, bounded from below away from zero, there exists a bounded open set $\Lambda \subset \Omega$ such that

$$\begin{aligned} \inf _{x\in \Lambda }V(x)<\min _{x\in \partial \Lambda }V(x), \end{aligned}$$

(1.5)

and the nonlinearity f satisfies some subcritical growth conditions. In [1], Alves and Figueiredo considered the following quasilinear elliptic equation

$$\begin{aligned} {\left\{ \begin{array}{ll} -\varepsilon ^{p}\Delta _{p} u+V(x)\vert u\vert ^{p-2}u=f(u) \hbox { in }\,{\mathbb {R}}^N,\\ u>0 \hbox { in }\,{\mathbb {R}}^N, \end{array}\right. } \end{aligned}$$

where V is a positive continuous function and satisfies the local assumption (1.5), $f\in C^{1}$ is a function having subcritical and superlinear growth. By using the Nehari manifold method and the Ljusternik–Schnirelmann category theory, the authors obtained the multiplicity of positive solutions. In order to apply the Nehari manifold method, the authors assumed that $f\in C^{1}$, which ensures that the Nehari manifold is a $C^{1}$-manifold. If f is only continuous, then the Nehari manifold is only a topological manifold, thus the arguments developed in [1] collapse. We notice that Szulkin and Weth in [34] considered the multiple solutions for the nonlinear stationary Schrödinger equation $-\Delta u+V(x) u=f(x, u)$ in ${\mathbb {R}}^N$, here f is superlinear, subcritical and continuous. In order to use the method of Nehari manifold, they developed a new approach. For further results about the existence, multiplicity and qualitative properties of semiclassical states with various types of concentration behaviors, which have been established under various assumptions on the potential V and on the nonlinearity f, see [2, 4, 6,7,8, 12, 14, 15, 19, 30, 31, 36] the references therein (see also [5, 23] for the fractional case).

On the other hand, the magnetic nonlinear Schrödinger equation (1.4) has been extensively investigated by many authors applying suitable variational and topological methods (see [3, 10, 11, 16,17,18, 21, 25, 26, 28] and references therein). To the best of our knowledge, the first result involving the magnetic field was obtained by Esteban and Lions [21]. They used the concentration-compactness principle and minimization arguments to obtain solutions for $\varepsilon >0$ fixed and $N=2, 3$. In particular, due to our scope, we want to mention [3] where the authors used the method of the Nehari manifold, the penalization method, and the Ljusternik–Schnirelmann category theory for a subcritical nonlinearity $f\in C^{1}$. We point out that if f is only continuous, then the arguments developed in [3] fail. Moreover, as we will see later, due to the presence of the magnetic field A(x), problem (1.4) cannot be changed into a pure real-valued problem, hence we must deal directly with a complex-valued problem, which causes several new difficulties in employing the methods to deal with our problem. Our problem is more complicated than the problem without magnetic field and we need additional technical estimates.

1.2 Main result

In this paper, motivated by [3, 24, 34], for the case where f is only continuous, we establish multiplicity and concentration properties of nontrivial solutions to problem (1.4).

Throughout the paper, we make the following assumptions on the potential V.

(V1):

There exists $V_{0}>0$ such that $V(x)\ge V_{0}$ for all $x\in {\mathbb {R}}^{N}$;

(V2):

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

$$\begin{aligned} V_{0}=\min _{x\in \Lambda }V(x)<\min _{x\in \partial \Lambda }V(x). \end{aligned}$$

Observe that

$$\begin{aligned} M:=\{x\in \Lambda : V(x)=V_{0}\}\ne \emptyset . \end{aligned}$$

Moreover, let the nonlinearity $f\in C({\mathbb {R}}, {\mathbb {R}})$ be a function satisfying the following hypotheses.

(f1):

$f(t)=0$ if $t\le 0$;

(f2):

There exists $q\in (2, 2^{*})$ such that

$$\begin{aligned} \lim _{t\rightarrow +\infty } \frac{f(t^{2})t}{t^{q-1}} =0, \end{aligned}$$

where $2^{*}=2N/(N-2)$ if $N\ge 3$, and $2^{*}=\infty$ if $N=2$;

(f3):

There is a positive constant $\theta >2$ such that

$$\begin{aligned} 0<\frac{\theta }{2} F(t)\le tf(t), \quad \quad \forall \, t>0, \quad \hbox {where } F(t)=\int _{0}^{t}f(s)ds; \end{aligned}$$

(f4):

f(t) is strictly increasing in $(0, \infty )$.

The main result of this paper is the following.

Theorem 1.1

Assume that V satisfies (V1), (V2) and f satisfies (f1)–(f4). Then, for any $\delta >0$ such that

$$\begin{aligned} M_{\delta }:=\{x\in {\mathbb {R}}^{N}: \mathrm{dist}\,(x, M)<\delta \}\subset \Lambda , \end{aligned}$$

there exists $\varepsilon _{\delta }>0$ such that, for any $0<\varepsilon <\varepsilon _{\delta }$, problem (1.4) has at least $\text {cat}_{M_{\delta }}(M)$ nontrivial solutions. Moreover, for every sequence $\{\varepsilon _n\}$ such that $\varepsilon _{n}\rightarrow 0^{+}$ as $n\rightarrow +\infty$, if we denote by $u_{\varepsilon _{n}}$ one of these solutions of problem (1.4) for $\varepsilon =\varepsilon _{n}$ and $\eta _{\varepsilon _{n}}\in {\mathbb {R}}^{N}$ is the global maximum point of $\vert u_{\varepsilon _{n}}\vert$, then

$$\begin{aligned} \lim _{n}V(\eta _{\varepsilon _{n}})=V_{0}. \end{aligned}$$

The paper is organized as follows. In Sect. 2, we introduce the functional setting and give some preliminaries. In Sect. 3, we study the modified problem. We prove the Palais-Smale condition for the modified energy functional and provide some tools which are useful to establish a multiplicity result. In Sect. 4, we study the autonomous problem associated. This allows us to show that the modified problem has multiple solutions. Finally, in Sect. 5, we complete the paper with the proof of Theorem 1.1. We refer to the recent monograph [32] for some of the main abstract methods used in this paper.

1.3 Notation

• $C, C_1, C_2, \ldots$ denote positive constants whose exact values are inessential and can change from line to line;

• $B_{R}(y)$ denotes the open disk centered at $y\in {\mathbb {R}}^N$ with radius $R>0$ and $B^{c}_{R}(y)$ denotes the complement of $B_{R}(y)$ in ${\mathbb {R}}^{N}$;

• $\Vert \cdot \Vert$, $\Vert \cdot \Vert _{q}$, and $\Vert \cdot \Vert _{L^{\infty }(\Omega )}$ denote the usual norms of the spaces $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, $L^{q}({\mathbb {R}}^{N}, {\mathbb {R}})$, and $L^{\infty }(\Omega , {\mathbb {R}})$, respectively, where $\Omega \subset {\mathbb {R}}^{N}$. $\langle \cdot , \cdot \rangle _{0}$ denotes the inner product of the space $H^{1}(\mathbb {R}^{N}, \mathbb {R})$.

2 Abstract setting and preliminary results

In this section, we present the functional spaces and some useful preliminary remarks which will be useful for our arguments. We also introduce a classical equivalent version of problem (1.4).

For $u: {\mathbb {R}}^{N}\rightarrow {\mathbb {C}}$, let us denote by

$$\begin{aligned} \nabla _{A}u:=\left( \frac{\nabla }{i}-A\right) u, \end{aligned}$$

and

$$\begin{aligned} H_{A}^{1}({\mathbb {R}}^{N}, {\mathbb {C}}):=\{u\in L^{2}({\mathbb {R}}^{N}, {\mathbb {C}}): |\nabla _{A}u|\in L^{2}({\mathbb {R}}^{N}, {\mathbb {R}})\}. \end{aligned}$$

The space $H_{A}^{1}({\mathbb {R}}^{N}, {\mathbb {C}})$ is an Hilbert space endowed with the scalar product

$$\begin{aligned} \langle u, v\rangle :={\text {Re}}\int _{{\mathbb {R}}^{2}}\Big (\nabla _{A}u\overline{\nabla _{A}v}+u{\overline{v}}\Big )dx,\quad \text {for any } u, v\in H_{A}^{1}({\mathbb {R}}^{N}, {\mathbb {C}}), \end{aligned}$$

where ${\text {Re}}$ and the bar denote the real part of a complex number and the complex conjugation, respectively. Moreover we denote by $\Vert u\Vert _{A}$ the norm induced by this inner product.

On $H_{A}^{1}({\mathbb {R}}^{N}, {\mathbb {C}})$ we will frequently use the following diamagnetic inequality (see, e.g., Lieb and Loss [29, Theorem 7.21])

$$\begin{aligned} \vert \nabla _{A}u(x)\vert \ge \vert \nabla \vert u(x)\vert \vert . \end{aligned}$$

(2.1)

Moreover, making a simple change of variables, we can see that (1.4) is equivalent to

$$\begin{aligned} \Big (\frac{1}{i}\nabla -A_{\varepsilon }(x)\Big )^{2}u+V_{\varepsilon }(x)u=f(\vert u\vert ^{2})u \quad \hbox {in }{\mathbb {R}}^N, \end{aligned}$$

(2.2)

where $A_{\varepsilon }(x)=A(\varepsilon x)$ and $V_{\varepsilon }(x)=V(\varepsilon x)$.

Let $H_{\varepsilon }$ be the Hilbert space obtained as the closure of $C_{c}^{\infty }({\mathbb {R}}^{N}, {\mathbb {C}})$ with respect to the scalar product

$$\begin{aligned} \langle u, v\rangle _{\epsilon } :={\text {Re}}\int _{{\mathbb {R}}^{N}}\Big (\nabla _{A_{\varepsilon }}u\overline{\nabla _{A_{\varepsilon }}v}+V_{\varepsilon }(x)u{\overline{v}}\Big )dx \end{aligned}$$

and let us denote by $\Vert \cdot \Vert _{\varepsilon }$ the norm induced by this inner product.

The diamagnetic inequality (2.1) implies that if $u\in H_{A_{\varepsilon }}^{1}({\mathbb {R}}^{N}, {\mathbb {C}})$, then $|u|\in H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$ and $\Vert u\Vert \le C \Vert u\Vert _\varepsilon$. Therefore, the embedding $H_{\varepsilon }\hookrightarrow L^{r}({\mathbb {R}}^{N}, {\mathbb {C}})$ is continuous for $2\le r\le 2^{*}$ and the embedding $H_{\varepsilon }\hookrightarrow L^{r}_{\text {loc}}({\mathbb {R}}^{N}, {\mathbb {C}})$ is compact for $1\le r<2^{*}$.

3 The modified problem

As in [20], to study system (1.4), or equivalently, problem (2.2) by variational methods, we modify suitably the nonlinearity f so that, for $\varepsilon >0$ small enough, the solutions of the modified problem are also solutions of the original one. More precisely, we choose $K>2$. By (f4) there exists a unique number $a>0$ verifying $K f(a)=V_{0}$, where $V_{0}$ is given in (V1). Hence we consider the function

$$\begin{aligned} {\tilde{f}}(t):= {\left\{ \begin{array}{ll} f(t), &{}\quad t\le a,\\ V_{0}/K, &{}\quad t> a. \end{array}\right. } \end{aligned}$$

Now we introduce the penalized nonlinearity $g:{\mathbb {R}}^{N}\times {\mathbb {R}}\rightarrow {\mathbb {R}}$

$$\begin{aligned} g(x, t):=\chi _{\Lambda }(x)f(t)+(1-\chi _{\Lambda }(x)){\tilde{f}}(t), \end{aligned}$$

(3.1)

where $\chi _{\Lambda }$ is the characteristic function on $\Lambda$ and $\displaystyle G(x, t):=\int _{0}^{t}g(x, s)ds$.

In view of (f1)–(f4), we deduce that g is a Carathéodory function satisfying the following properties:

(g1):

$g(x, t)=0$ for each $t\le 0$;

(g2):

$\underset{t\rightarrow 0^{+}}{\lim }g(x, t)=0$ uniformly in $x\in {\mathbb {R}}^{N}$;

(g3):

$g(x, t)\le f(t)$ for all $t\ge 0$ and uniformly in $x\in {\mathbb {R}}^{N}$;

(g4):

$0<\theta G(x, t)\le 2 g(x, t)t$, for each $x\in \Lambda$, $t>0$;

(g5):

$0< G(x, t)\le g(x, t)t\le V_{0}t/K$, for each $x\in \Lambda ^{c}$, $t>0$;

(g6):

for each $x\in \Lambda$, the function $t\mapsto g(x, t)$ is strictly increasing in $t\in (0, +\infty )$ and for each $x\in \Lambda ^c$, the function $t\mapsto g(x, t)$ is strictly increasing in (0, a).

Then we consider the modified problem

$$\begin{aligned} \Big (\frac{1}{i}\nabla -A_{\varepsilon }(x)\Big )^{2}u +V_{\varepsilon }(x)u=g(\varepsilon x, \vert u\vert ^{2})u \quad \hbox {in }{\mathbb {R}}^N. \end{aligned}$$

(3.2)

Note that if u is a solution of problem (3.2) with

$$\begin{aligned} \vert u(x)\vert ^2 \le a\quad \text {for all } x\in \Lambda ^{c}_{\varepsilon }, \quad \Lambda _{\varepsilon }:=\{x\in {\mathbb {R}}^{N}: \varepsilon x\in \Lambda \}, \end{aligned}$$

then u is a solution of problem (2.2).

The energy functional associated to problem (3.2) is

$$\begin{aligned} J_{\varepsilon }(u) :=\frac{1}{2}\int _{{\mathbb {R}}^{N}}(\vert \nabla _{A_\varepsilon }u\vert ^{2}+V_\varepsilon (x)\vert u\vert ^{2})dx -\frac{1}{2}\int _{{\mathbb {R}}^{N}} G(\varepsilon x, \vert u\vert ^{2})dx\quad \text{ for } \text{ all } u\in H_{\varepsilon }. \end{aligned}$$

It is standard to prove that $J_{\varepsilon }\in C^{1}(H_{\varepsilon }, {\mathbb {R}})$ and its critical points are the weak solutions of the modified problem (3.2).

We denote by ${\mathcal {N}}_{\varepsilon }$ the Nehari manifold of $J_{\varepsilon }$, that is,

$$\begin{aligned} {\mathcal {N}}_{\varepsilon }:=\{u\in H_{\varepsilon }\backslash \{0\}: J'_{\varepsilon }(u)[u]=0\}, \end{aligned}$$

and define the number $c_{\varepsilon }$ by

$$\begin{aligned} c_{\varepsilon }=\inf _{u\in {\mathcal {N}}_{\varepsilon }}J_{\varepsilon }(u). \end{aligned}$$

Let $H_{\varepsilon }^{+}$ be the open subset of $H_{\varepsilon }$ given by

$$\begin{aligned} H_{\varepsilon }^{+}=\{u\in H_{\varepsilon }: \vert \text {supp}(u)\cap \Lambda _{\varepsilon }\vert >0\}, \end{aligned}$$

and $S_{\varepsilon }^{+}=S_{\varepsilon }\cap H_{\varepsilon }^{+}$, where $S_{\varepsilon }$ is the unit sphere of $H_{\varepsilon }$. Note that $S_{\varepsilon }^{+}$ is a non-complete $C^{1, 1}$-manifold of codimension 1, modeled on $H_{\varepsilon }$ and contained in $H_{\varepsilon }^{+}$. Therefore, $H_{\varepsilon }=T_{u}S_{\varepsilon }^{+}\bigoplus {\mathbb {R}}u$ for each $u\in T_{u}S_{\varepsilon }^{+}$, where $T_{u}S_{\varepsilon }^{+}=\{v\in H_{\varepsilon }: \langle u, v\rangle _{\epsilon }=0\}$.

Now we show that the functional $J_{\varepsilon }$ satisfies the mountain pass geometry (see [9, 33, 37]).

Lemma 3.1

For any fixed $\varepsilon >0$, the functional $J_{\varepsilon }$ satisfies the following properties:

• (i) there exist $\beta , r>0$ such that $J_{\varepsilon }(u)\ge \beta$ if $\Vert u\Vert _{\varepsilon }=r$;

• (ii) there exists $e\in H_{\varepsilon }$ with $\Vert e\Vert _{\varepsilon }>r$ such that $J_{\varepsilon }(e)<0$.

Proof

(i)   By (g3), (f1) and (f2), for any $\zeta >0$ small, there exists $C_{\zeta }>0$ such that

$$\begin{aligned} G(\varepsilon x, \vert u\vert ^{2})\le \zeta \vert u\vert ^{2}+C_{\zeta }\vert u\vert ^{q}\quad \text {for all } x\in {\mathbb {R}}^{N}. \end{aligned}$$

By the Sobolev embedding it follows that

$$\begin{aligned} J_{\varepsilon }(u)&\ge \frac{1}{2}\int _{{\mathbb {R}}^{N}}(\vert \nabla _{A_\varepsilon }u\vert ^{2}+V_\varepsilon (x)\vert u\vert ^{2})dx-\frac{\zeta }{2}\int _{{\mathbb {R}}^{N}}\vert u\vert ^{2}dx-\frac{C_{\zeta }}{2}\int _{{\mathbb {R}}^{N}}\vert u\vert ^{q}dx\\&\ge \frac{1}{4}\Vert u_{n}\Vert ^{2}_{\varepsilon }-CC_{\zeta }\Vert u_{n}\Vert ^{q}_{\varepsilon }. \end{aligned}$$

Hence we can choose some $\beta , r>0$ such that $J_{\varepsilon }(u)\ge \beta$ if $\Vert u\Vert _{\varepsilon }=r$ since $q>2$.

(ii)   For each $u\in H_{\varepsilon }^{+}$ and $t>0$, by the definition of g and (f3), one has

$$\begin{aligned} J_{\varepsilon }(t u)&\le \frac{t^{2}}{2}\int _{{\mathbb {R}}^{N}}(\vert \nabla _{A_\varepsilon }u\vert ^{2}+V_\varepsilon (x)\vert u\vert ^{2})dx -\frac{1}{2}\int _{\Lambda _{\varepsilon }} G(\varepsilon x, t^{2}\vert u\vert ^{2})dx,\\&\le \frac{t^{2}}{2}\Vert u\Vert _{\varepsilon }^{2} -C_{1}t^{\theta }\int _{\Lambda _{\varepsilon }} \vert u\vert ^{\theta }dx+C_{2}\vert \text {supp}(u)\cap \Lambda _{\varepsilon }\vert . \end{aligned}$$

Since $\theta >2$, we get the conclusion. $\square$

Since f is only continuous, the next results are very important because they allow us to overcome the non-differentiability of ${\mathcal {N}}_{\varepsilon }$ and the incompleteness of $S_{\varepsilon }^{+}$.

Lemma 3.2

Assume that (V1)–(V2) and (f1)–(f4) are satisfied, then the following properties hold.

(A1):

For any $u\in H_{\varepsilon }^{+}$, let $g_{u}: {\mathbb {R}}^{+}\rightarrow {\mathbb {R}}$ be given by $g_{u}(t)=J_{\varepsilon }(t u)$. Then there exists a unique $t_{u}>0$ such that $g'_{u}(t)>0$ in $(0, t_{u})$ and $g'_{u}(t)<0$ in $(t_{u}, \infty )$.

(A2):

There exists $\tau >0$ independent on u such that $t_{u}\ge \tau$ for all $u\in S_{\varepsilon }^{+}$. Moreover, for each compact ${\mathcal {W}}\subset S_{\varepsilon }^{+}$ there is $C_{\mathcal {W}}$ such that $t_{u}\le C_{{\mathcal {W}}}$, for all $u\in {\mathcal {W}}$.

(A3):

The map ${\widehat{m}}_{\varepsilon }: H_{\varepsilon }^{+}\rightarrow {\mathcal {N}}_{\varepsilon }$ given by ${\widehat{m}}_{\varepsilon }(u)=t_{u}u$ is continuous and $m_{\varepsilon }={\widehat{m}}_{\varepsilon }|_{S_{\varepsilon }^{+}}$ is a homeomorphism between $S_{\varepsilon }^{+}$ and ${\mathcal {N}}_{\varepsilon }$. Moreover, $m_{\varepsilon }^{-1}(u)=\frac{u}{\Vert u\Vert _{\varepsilon }}$.

(A4):

If there is a sequence $\{u_{n}\}\subset S_{\varepsilon }^{+}$ such that $\text {dist}(u_{n}, \partial S_{\varepsilon }^{+})\rightarrow 0$, then $\Vert m_{\varepsilon }(u_{n})\Vert _{\varepsilon }\rightarrow \infty$ and $J_{\varepsilon }(m_{\varepsilon }(u_{n}))\rightarrow \infty$.

Proof

(A1) As in the proof of Lemma 3.1, we have $g_{u}(0)=0$, $g_{u}(t)>0$ for $t>0$ small and $g_{u}(t)<0$ for $t>0$ large. Therefore, $\max _{t\ge 0}g_{u}(t)$ is achieved at a global maximum point $t=t_{u}$ verifying $g'_{u}(t_{u})=0$ and $t_{u}u\in {\mathcal {N}}_{\varepsilon }$. From (f4), the definition of g and $\vert \text {supp}(u)\cap \Lambda _{\varepsilon }\vert >0$, we may obtain the uniqueness of $t_{u}$. Therefore, $\max _{t\ge 0}g_{u}(t)$ is achieved at a unique $t=t_{u}$ so that $g'_{u}(t)=0$ and $t_{u}u\in {\mathcal {N}}_{\varepsilon }$.

(A2) For $\forall \,u\in S_{\varepsilon }^{+}$, we have

$$\begin{aligned} t_{u} =\int _{{\mathbb {R}}^{N}} g(\varepsilon x, t_{u}^{2}\vert u\vert ^{2})t_{u}\vert u\vert ^{2} dx. \end{aligned}$$

From (g2), (g3), the Sobolev embeddings and $q>2$, we get

$$\begin{aligned} t_{u} \le \zeta t_{u}\int _{{\mathbb {R}}^{N}}\vert u\vert ^{2}dx+C_{\zeta }t_{u}^{q-1}\int _{{\mathbb {R}}^{N}}\vert u\vert ^{q}dx\le C_{1}\zeta t_{u}+C_{2}C_{\zeta }t_{u}^{q-1}, \end{aligned}$$

which implies that $t_{u}\ge \tau$ for some $\tau >0$. If ${\mathcal {W}}\subset S_{\varepsilon }^{+}$ is compact, and suppose by contradiction that there is $\{u_{n}\}\subset {\mathcal {W}}$ with $t_{n}:=t_{u_{n}}\rightarrow \infty$. Since ${\mathcal {W}}$ is compact, there exists $u\in {\mathcal {W}}$ such that $u_{n}\rightarrow u$ in $H_{\varepsilon }$. Moreover, using the proof of Lemma 3.1(ii), we have that $J_{\varepsilon }(t_{n} u_{n})\rightarrow -\infty$.

On the other hand, let $v_{n}: =t_{n}u_{n}\in {\mathcal {N}}_{\varepsilon }$, from (g4), (g5) and (g6), it yields that

$$\begin{aligned} J_{\varepsilon }(v_{n})&= J_{\varepsilon }(v_{n}) -\frac{1}{\theta }J_{\varepsilon }'(v_{n})[v_{n}]\\&\ge \Big (\frac{1}{2}-\frac{1}{\theta }\Big )\Vert v_{n}\Vert ^{2}_{\varepsilon } +\int _{\Lambda _{\varepsilon }^{c}} \Big (\frac{1}{\theta }g(\varepsilon x, \vert v_{n}\vert ^{2})\vert v_{n}\vert ^{2}-\frac{1}{2}G(\varepsilon x, \vert v_{n}\vert ^{2})\Big )dx\\&\ge \Big (\frac{1}{2}-\frac{1}{\theta }\Big )\Big (\Vert v_{n}\Vert ^{2}_{\varepsilon }-\frac{1}{K}\int _{{\mathbb {R}}^{N}} V(\varepsilon x)\vert v_{n}\vert ^{2}dx\Big )\\&\ge \Big (\frac{1}{2}-\frac{1}{\theta }\Big )\left( 1-\frac{1}{K}\right) \Vert v_{n}\Vert ^{2}_{\varepsilon }. \end{aligned}$$

Thus, substituting $v_{n}: =t_{n}u_{n}$ and $\Vert v_{n}\Vert _{\varepsilon }=t_{n}$, we obtain

$$\begin{aligned} 0<\Big (\frac{1}{2}-\frac{1}{\theta }\Big )\left( 1-\frac{1}{K}\right) \le \frac{J_{\varepsilon }(v_{n})}{t_{n}^{2}}\le 0 \end{aligned}$$

as $n\rightarrow \infty$, which yields a contradiction. This proves (A2).

(A3) First of all, we note that ${\widehat{m}}_{\varepsilon }$, $m_{\varepsilon }$ and $m_{\varepsilon }^{-1}$ are well defined. Indeed, by (A2), for each $u\in H_{\varepsilon }^{+}$, there is a unique ${\widehat{m}}_{\varepsilon }(u)\in {\mathcal {N}}_{\varepsilon }$. On the other hand, if $u\in {\mathcal {N}}_{\varepsilon }$, then $u\in H_{\varepsilon }^{+}$. Otherwise, we have $\vert \text {supp}(u)\cap \Lambda _{\varepsilon }\vert =0$ and by (g5) we have

$$\begin{aligned} \Vert u\Vert ^{2}_{\varepsilon }&=\int _{{\mathbb {R}}^{N}}g(\varepsilon x, \vert u\vert ^{2})\vert u\vert ^{2}dx=\int _{\Lambda _{\varepsilon }^{c}} g(\varepsilon x, \vert u\vert ^{2})\vert u\vert ^{2}dx\le \frac{1}{K}\int _{{\mathbb {R}}^{N}} V(\varepsilon x)\vert u\vert ^{2}dx\\&\le \frac{1}{K}\Vert u\Vert ^{2}_{\varepsilon } \end{aligned}$$

which is impossible since $K>1$ and $u\ne 0$. Therefore, $m_{\varepsilon }^{-1}(u)=\frac{u}{\Vert u\Vert _{\varepsilon }}\in S_{\varepsilon }^{+}$ is well defined and continuous. From

$$\begin{aligned} m_{\varepsilon }^{-1}\big (m_{\varepsilon }(u)\big )=m_{\varepsilon }^{-1}(t_{u}u)=\frac{t_{u}u}{t_{u}\Vert u\Vert _{\varepsilon }}=u,\,\,\,\forall u\in S_{\varepsilon }^{+}, \end{aligned}$$

we conclude that $m_{\varepsilon }$ is a bijection. Now we prove ${\widehat{m}}_{\varepsilon }: H_{\varepsilon }^{+}\rightarrow {\mathcal {N}}_{\varepsilon }$ is continuous, let $\{u_{n}\}\subset H_{\varepsilon }^{+}$ and $u\in H_{\varepsilon }^{+}$ such that $u_{n}\rightarrow u$ in $H_{\varepsilon }$. By (A2), there is a $t_{0}>0$ such that $t_{n}:=t_{u_{n}}\rightarrow t_{0}$. Using $t_{n}u_{n}\in {\mathcal {N}}_{\varepsilon }$, i.e.,

$$\begin{aligned} t_{n}^{2}\Vert u_{n}\Vert ^{2}_{\varepsilon } =\int _{{\mathbb {R}}^{N}} g(\varepsilon x, t_{n}^{2}\vert u_{n}\vert ^{2})t_{n}^{2}\vert u_{n}\vert ^{2} dx,\,\,\,\forall n\in N, \end{aligned}$$

and passing to the limit as $n\rightarrow \infty$ in the last equality, we obtain

$$\begin{aligned} t_{0}^{2}\Vert u\Vert ^{2}_{\varepsilon } =\int _{{\mathbb {R}}^{N}} g(\varepsilon x, t_{0}^{2}\vert u\vert ^{2})t_{0}^{2}\vert u\vert ^{2} dx, \end{aligned}$$

which implies that $t_{0}u\in {\mathcal {N}}_{\varepsilon }$ and $t_{u}= t_{0}$. This proves ${\widehat{m}}_{\varepsilon }(u_{n})\rightarrow {\widehat{m}}_{\varepsilon }(u)$ in $H_{\varepsilon }^{+}$. Thus, ${\widehat{m}}_{\varepsilon }$ and $m_{\varepsilon }$ are continuous functions and (A3) is proved.

(A4)   Let $\{u_{n}\}\subset S_{\varepsilon }^{+}$ be a subsequence such that $\text {dist}(u_{n}, \partial S_{\varepsilon }^{+})\rightarrow 0$, then for each $v\in \partial S_{\varepsilon }^{+}$ and $n\in N$, we have $\vert u_{n}\vert = \vert u_{n}-v\vert$ a.e. in $\Lambda _{\varepsilon }$. Therefore, by (V1), (V2) and the Sobolev embedding, there exists a constant $C_{t}>0$ such that

$$\begin{aligned} \Vert u_n\Vert _{L^{t}(\Lambda _{\varepsilon })}&\le \inf _{v\in \partial S_{\varepsilon }^{+}}\Vert u_n-v\Vert _{L^{t}(\Lambda _{\varepsilon })}\\&\le C_{t}\left( \inf _{v\in \partial S_{\varepsilon }^{+}}\int _{\Lambda _{\varepsilon }}(\vert \nabla _{A_\varepsilon }u_n-v\vert ^{2}+V_\varepsilon (x)\vert u_n-v\vert ^{2})dx\right) ^{\frac{1}{2}}\\&\le C_{t}\,\text {dist}(u_{n}, \partial S_{\varepsilon }^{+}) \end{aligned}$$

for all $n\in N$, $t\in [2, 2^{*}]$. By (g2), (g3) and (g5), for each $t>0$, we have

$$\begin{aligned} \int _{{\mathbb {R}}^{N}} G(\varepsilon x, t^{2}\vert u_{n}\vert ^{2})dx&\le \int _{\Lambda _{\varepsilon }} F( t^{2}\vert u_{n}\vert ^{2})dx+\frac{t^{2}}{K}\int _{\Lambda _{\varepsilon }^{c}} V(\varepsilon x)\vert u_{n}\vert ^{2}dx\\&\le C_{1}t^{2} \int _{\Lambda _{\varepsilon }}\vert u_{n}\vert ^{2}dx+C_{2}t^{q} \int _{\Lambda _{\varepsilon }}\vert u_{n}\vert ^{q}dx +\frac{t^{2}}{K}\Vert u_{n}\Vert ^{2}_{\varepsilon }\\&\le C_{3}t^{2} \text {dist}(u_{n}, \partial S_{\varepsilon }^{+})^{2}+C_{4}t^{q} \text {dist}(u_{n}, \partial S_{\varepsilon }^{+})^{q} +\frac{t^{2}}{K}. \end{aligned}$$

Therefore,

$$\begin{aligned} \limsup _{n}\int _{{\mathbb {R}}^{N}} G(\varepsilon x, t^{2}\vert u_{n}\vert ^{2})dx\le \frac{t^{2}}{K},\,\,\forall t>0. \end{aligned}$$

On the other hand, from the definition of $m_{\varepsilon }$ and the last inequality, for all $t>0$, one has

$$\begin{aligned} \liminf _{n}J_{\varepsilon }(m_{\varepsilon }(u_{n}))&\ge \liminf _{n}J_{\varepsilon }(t u_{n})\\&\ge \lim \inf _{n} \frac{t^{2}}{2}\Vert u_{n}\Vert ^{2}_{\varepsilon }-\frac{t^{2}}{K}\\&=\frac{K-2}{2K}t^{2}, \end{aligned}$$

this implies that

$$\begin{aligned} \liminf _{n}\frac{1}{2}\Vert m_{\varepsilon }(u_{n})\Vert _{\varepsilon }^{2}\ge \liminf _{n}J_{\varepsilon }(m_{\varepsilon }(u_{n}))\ge \frac{K-2}{2K}t^{2},\,\, \forall \, t>0. \end{aligned}$$

From the arbitrariness of $t>0$, it is easy to see that $\Vert m_{\varepsilon }(u_{n})\Vert _{\varepsilon }\rightarrow \infty$ and $J_{\varepsilon }(m_{\varepsilon }(u_{n}))\rightarrow \infty$ as $n\rightarrow \infty$. This completes the proof of Lemma 3.2. $\square$

Now we define the function

$$\begin{aligned} {\widehat{\Psi }}_{\varepsilon }: H_{\varepsilon }^{+}\rightarrow {\mathbb {R}}, \end{aligned}$$

by ${\widehat{\Psi }}_{\varepsilon }(u)=J_{\varepsilon }({\widehat{m}}_{\varepsilon }(u))$ and denote by $\Psi _{\varepsilon }:=({\widehat{\Psi }}_{\varepsilon })|_{S_{\varepsilon }^{+}}$.

From Lemma 3.2, arguing as Corollary 10 in [35], we may obtain the following result.

Lemma 3.3

Assume that (V1)–(V2) and (f1)–(f4) are satisfied, then

(B1):

${\widehat{\Psi }}_{\varepsilon }\in C^{1}(H_{\varepsilon }^{+}, {\mathbb {R}})$ and

$$\begin{aligned} {\widehat{\Psi }}_{\varepsilon }'(u)v=\frac{\Vert {\widehat{m}}_{\varepsilon }(u)\Vert _{\epsilon }}{\Vert u\Vert _{\epsilon }}J'_{\varepsilon }({\widehat{m}}_{\varepsilon }(u))[v],\,\,\,\forall \, u\in H_{\varepsilon }^{+}\,\,\text {and}\,\,\forall \, v\in H_{\varepsilon }; \end{aligned}$$

(B2):

$\Psi _{\varepsilon }\in C^{1}(S_{\varepsilon }^{+}, {\mathbb {R}})$ and

$$\begin{aligned} \Psi _{\varepsilon }'(u)v=\Vert m_{\varepsilon }(u)\Vert _{\epsilon }J'_{\varepsilon }({\widehat{m}}_{\varepsilon }(u))[v],\,\,\,\forall v\in T_{u}S_{\varepsilon }^{+}; \end{aligned}$$

(B3):

If $\{u_{n}\}$ is a $(PS)_{c}$ sequence of $\Psi _{\varepsilon }$, then $\{m_{\varepsilon }(u_{n})\}$ is a $(PS)_{c}$ sequence of $J_{\varepsilon }$. If $\{u_{n}\}\subset {\mathcal {N}}_{\varepsilon }$ is a bounded $(PS)_{c}$ sequence of $J_{\varepsilon }$, then $\{m_{\varepsilon }^{-1}(u_{n})\}$ is a $(PS)_{c}$ sequence of $\Psi _{\varepsilon }$;

(B4):

u is a critical point of $\Psi _{\varepsilon }$ if and only if $m_{\varepsilon }(u)$ is a critical point of $J_{\varepsilon }$. Moreover, the corresponding critical values coincide and

$$\begin{aligned} \inf _{S_{\varepsilon }^{+}}\Psi _{\varepsilon }.=\inf _{{\mathcal {N}}_{\varepsilon }}J_{\varepsilon }. \end{aligned}$$

As in [35], we have the following variational characterization of the infimum of $J_{\varepsilon }$ over ${\mathcal {N}}_{\varepsilon }$:

$$\begin{aligned} c_{\varepsilon }=\inf _{u\in {\mathcal {N}}_{\varepsilon }}J_{\varepsilon }(u)=\inf _{u\in H_{\varepsilon }^{+}}\sup _{t> 0}J_{\varepsilon }(tu)=\inf _{u\in S_{\varepsilon }^{+}}\sup _{t> 0}J_{\varepsilon }(tu). \end{aligned}$$

Lemma 3.4

Let $c>0$ and $\{u_{n}\}$ is a $(PS)_{c}$ sequence for $J_{\varepsilon }$, then $\{u_{n}\}$ is bounded in $H_{\varepsilon }$.

Proof

Assume that $\{u_{n}\}\subset H_{\varepsilon }$ is a $(PS)_{c}$ sequence for $J_{\varepsilon }$, that is, $J_{\varepsilon }(u_{n})\rightarrow c$ and $J'_{\varepsilon }(u_{n})\rightarrow 0$. By using (g4) and (g5), we have

$$\begin{aligned} c+o_{n}(1)+o_{n}(1)\Vert u_n\Vert _\varepsilon&\ge J_{\varepsilon }(u_{n}) -\frac{1}{\theta }J_{\varepsilon }'(u_{n})[u_{n}]\\&= \Big (\frac{1}{2}-\frac{1}{\theta }\Big )\Vert u_{n}\Vert ^{2}_{\varepsilon } + \int _{{\mathbb {R}}^{N}}\Big (\frac{1}{\theta }g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2}-\frac{1}{2}G(\varepsilon x, \vert u_{n}\vert ^{2})\Big )dx\\&\ge \Big (\frac{1}{2}-\frac{1}{\theta }\Big )\Vert u_{n}\Vert ^{2}_{\varepsilon } +\int _{\Lambda _{\varepsilon }^{c}} \Big (\frac{1}{\theta }g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2}-\frac{1}{2}G(\varepsilon x, \vert u_{n}\vert ^{2})\Big )dx\\&\ge \Big (\frac{1}{2}-\frac{1}{\theta }\Big )\Big (\Vert u_{n}\Vert ^{2}_{\varepsilon }-\int _{\Lambda _{\varepsilon }^{c}} g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2} dx\Big )\\&\ge \Big (\frac{1}{2}-\frac{1}{\theta }\Big )\Big (\Vert u_{n}\Vert ^{2}_{\varepsilon }-\frac{1}{K}\int _{{\mathbb {R}}^{N}} V(\varepsilon x)\vert u_{n}\vert ^{2}dx\Big )\\&\ge \Big (\frac{1}{2}-\frac{1}{\theta }\Big )(1-\frac{1}{K})\Vert u_{n}\Vert ^{2}_{\varepsilon }. \end{aligned}$$

Since $K>2$, from the above inequalities we obtain that $\{u_{n}\}$ is bounded in $H_{\varepsilon }$. $\square$

The following result is important to prove the $(PS)_{c_{\varepsilon }}$ condition for the functional $J_{\varepsilon }$.

Lemma 3.5

The functional $J_{\varepsilon }$ satisfies the $(PS)_{c}$ condition at any level $c>0$.

Proof

Let $(u_{n})\subset H_{\varepsilon }$ be a $(PS)_{c}$ sequence for $J_{\varepsilon }$. By Lemma 3.4, $(u_{n})$ is bounded in $H_{\varepsilon }$. Thus, up to a subsequence, $u_{n}\rightharpoonup u$ in $H_{\varepsilon }$ and $u_{n}\rightarrow u$ in $L^{r}_{\mathrm{loc}}({\mathbb {R}}^{N}, {\mathbb {C}})$ for all $1\le r< 2^{*}$ as $n\rightarrow +\infty$. Moreover, the subcritical growth of g imply that $J'_{\varepsilon }(u)=0$, and

$$\begin{aligned} \Vert u\Vert _{\varepsilon }^{2}=\int _{{\mathbb {R}}^{N}}g(\varepsilon x, \vert u\vert ^{2})\vert u\vert ^{2} dx. \end{aligned}$$

Let $R>0$ be such that $\Lambda _{\varepsilon }\subset B_{R/2}(0)$. We show that for any given $\zeta >0$, for R large enough,

$$\begin{aligned} \limsup _n\int _{B_{R}^c(0)}(\vert \nabla _{A_\varepsilon }u_{n}\vert ^{2}+V_{\varepsilon }(x)\vert u_{n}\vert ^{2})dx\le \zeta . \end{aligned}$$

(3.3)

Let $\phi _{R}\in C^{\infty }({\mathbb {R}}^{N}, {\mathbb {R}})$ be a cut-off function such that

$$\begin{aligned} \phi _{R}=0\quad x\in B_{R/2}(0), \quad \phi _{R}=1\quad x\in B^{c}_{R}(0),\quad 0\le \phi _{R}\le 1,\quad \text {and}\, \quad \vert \nabla \phi _{R}\vert \le C/R \end{aligned}$$

where $C>0$ is a constant independent of R. Since the sequence $(\phi _{R}u_{n})$ is bounded in $H_{\varepsilon }$, we have

$$\begin{aligned} J_{\varepsilon }'( u_{n})[\phi _{R} u_{n}]=o_{n}(1), \end{aligned}$$

that is

$$\begin{aligned} {\text {Re}}\int _{{\mathbb {R}}^{N}}\nabla _{A_\varepsilon }u_{n}\overline{\nabla _{A_\varepsilon }(\phi _{R}u_{n})}dx+\int _{{\mathbb {R}}^{N}}V_{\varepsilon }(x)\vert u_{n}\vert ^{2}\phi _{R}dx=\int _{{\mathbb {R}}^{N}}g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2}\phi _{R}dx+o_{n}(1). \end{aligned}$$

Since $\overline{\nabla _{A_\varepsilon }(u_{n}\phi _{R})}=i\overline{u_{n}}\nabla \phi _{R}+\phi _{R}\overline{\nabla _{A_\varepsilon }u_{n}}$, using (g5), we have

$$\begin{aligned}&\int _{{\mathbb {R}}^{N}}(\vert \nabla _{A_\varepsilon }u_{n}\vert ^{2}+V_{\varepsilon }(x)\vert u_{n}\vert ^{2})\phi _{R}dx \\&\quad =\int _{{\mathbb {R}}^{N}}g( \varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2}\phi _{R}dx - {\text {Re}}\int _{{\mathbb {R}}^{N}}i\overline{u_{n}}\nabla _{A_\varepsilon }u_{n}\nabla \phi _{R}dx +o_{n}(1)\\&\quad \le \frac{1}{K}\int _{{\mathbb {R}}^{N}}V_{\varepsilon }(x)\vert u_{n}\vert ^{2}\phi _{R}dx - {\text {Re}}\int _{{\mathbb {R}}^{N}}i\overline{u_{n}}\nabla _{A_\varepsilon }u_{n}\nabla \phi _{R}dx +o_{n}(1). \end{aligned}$$

By the definition of $\phi _{R}$, the Hölder inequality and the boundedness of $(u_{n})$ in $H_{\varepsilon }$, we obtain

$$\begin{aligned} \Big (1-\frac{1}{K}\Big )\int _{{\mathbb {R}}^{N}}(\vert \nabla _{A_\varepsilon }u_{n}\vert ^{2}+V_{\varepsilon }(x)\vert u_{n}\vert ^{2})\phi _{R}dx\le \frac{C}{R}\Vert {u}_{n}\Vert _{2}\Vert \nabla _{A_\varepsilon }u_{n}\Vert _{2}+o_{n}(1)\le \frac{C_{1}}{R}+o_{n}(1) \end{aligned}$$

and so (3.3) holds.

Using $u_{n}\rightarrow u$ in $L_{\mathrm{loc}}^{r}({\mathbb {R}}^{N})$, for all $1\le r<2^{*}$ again, up to a subsequence, we have that

$$\begin{aligned} \vert u_{n}\vert \rightarrow \vert u\vert \text { a.e. in }{\mathbb {R}}^N\text { as }n\rightarrow +\infty , \end{aligned}$$

then

$$\begin{aligned} g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2}\rightarrow g(\varepsilon x, \vert u\vert ^{2})\vert u\vert ^{2} \text { a.e. in } {\mathbb {R}}^N\text { as }n\rightarrow +\infty . \end{aligned}$$

Moreover, from the subcritical growth of g and and the Lebesgue Dominated Convergence Theorem, we can infer

$$\begin{aligned} \lim _n \int _{B_{R}(0)}\Big |g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2} - g(\varepsilon x, \vert u\vert ^{2})\vert u\vert ^{2}\Big |dx=0. \end{aligned}$$

Now, by (g5) and (3.3) we have

$$\begin{aligned} \int _{B_{R}^c(0)}\Big \vert g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2}- g(\varepsilon x, \vert u\vert ^{2})\vert u\vert ^{2}\Big \vert dx \le \frac{2}{K}\int _{ B_{R}^c(0)}(\vert \nabla _{A_\varepsilon }u_{n}\vert ^{2}+V(\varepsilon x)\vert u_{n}\vert ^{2})dx < \frac{2\zeta }{K} \end{aligned}$$

for every $\zeta >0$.

Therefore

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2}dx\rightarrow \int _{{\mathbb {R}}^{N}}g(\varepsilon x, \vert u\vert ^{2})\vert u\vert ^{2}dx \text { as } n\rightarrow +\infty . \end{aligned}$$

Finally, since $J_{\varepsilon }'(u)=0$, we have

$$\begin{aligned} o_{n}(1) =J'_{\varepsilon }( u_{n}) [u_{n}] =\Vert u_{n}\Vert _{\varepsilon }^{2}-\int _{{\mathbb {R}}^{N}}g(\varepsilon x, \vert u_{n}\vert ^{2})\vert u_{n}\vert ^{2}dx =\Vert u_{n}\Vert _{\varepsilon }^{2}-\Vert u\Vert _{\varepsilon }^{2} + o_n(1). \end{aligned}$$

Thus, the sequence $(u_{n})$ strong converges to u in $H_{\varepsilon }$. $\square$

Since f is only assumed to be continuous, the following result is required for the multiplicity result in the next section.

Corollary 3.1

The functional $\Psi _{\varepsilon }$ satisfies the $(PS)_{c}$ condition on $S_{\varepsilon }^{+}$ at any level $c>0$.

Proof

Let $\{u_{n}\}\subset S_{\varepsilon }^{+}$ be a $(PS)_{c}$ sequence for $\Psi _{\varepsilon }$. Then $\Psi _{\varepsilon }(u_{n})\rightarrow c$ and $\Vert \Psi '_{\varepsilon }(u_{n})\Vert _{*}\rightarrow 0$, where $\Vert \cdot \Vert _{*}$ is the norm in the dual space $(T_{u_{n}}S_{\varepsilon }^{+})^{*}$. By Lemma 3.3(B3), we know that $\{m_{\varepsilon }(u_{n})\}$ is a $(PS)_{c}$ sequence for $J_{\varepsilon }$ in $H_{\varepsilon }$. From Lemma 3.5, we know that there exists a $u\in S_{\varepsilon }^{+}$ such that, up to a subsequence, $m_{\varepsilon }(u_{n})\rightarrow m_{\varepsilon }(u)$ in $H_{\varepsilon }$. By Lemma 3.2(A3), we obtain

$$\begin{aligned} u_{n}\rightarrow u\,\,\text {in}\,\,S_{\varepsilon }^{+}, \end{aligned}$$

and the proof is complete. $\square$

Proposition 3.1

Assume that (V1)–(V2) and (f1)–(f4) hold, then problem (3.2) has a ground state solution for any $\epsilon >0$.

Proof

Since

$$\begin{aligned} c_{\varepsilon }=\inf _{u\in {\mathcal {N}}_{\varepsilon }}J_{\varepsilon }(u)=\inf _{u\in H_{\varepsilon }^{+}}\sup _{t> 0}J_{\varepsilon }(tu)=\inf _{u\in S_{\varepsilon }^{+}}\sup _{t> 0}J_{\varepsilon }(tu), \end{aligned}$$

by the Ekeland variational principle [37], we obtain a minimizing $(PS)_{c_{\varepsilon }}$ sequence on $S_{\varepsilon }^{+}$ for the functional $\Psi _{\varepsilon }$. Moreover, by Corollary 3.1, we deduce the existence of a ground state $u\in H_{\varepsilon }$ for problem (3.2). $\square$

4 Multiple solutions for the modified problem

4.1 The autonomous problem

For our scope, we need also to study the following limit problem

$$\begin{aligned} -\Delta u+V_{0}u=f(u^{2})u, \quad u:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}, \end{aligned}$$

(4.1)

whose associated $C^1$-functional, defined in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, is

$$\begin{aligned} I_{0}(u):=\frac{1}{2}\int _{{\mathbb {R}}^{N}}(\vert \nabla u\vert ^{2}+V_{0} u^{2})dx -\frac{1}{2}\int _{{\mathbb {R}}^{N}}F(u^{2})dx. \end{aligned}$$

Let

$$\begin{aligned} {\mathcal {N}}_{0}:=\{u\in H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})\setminus \{0\}: I'_{0}(u)[u]=0\} \end{aligned}$$

and

$$\begin{aligned} c_{V_{0}}:=\inf _{u\in {\mathcal {N}}_{0}} I_{0}(u). \end{aligned}$$

Let $S_{0}$ be the unit sphere of $H_{0}:= H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$. Note that $S_{0}$ is a complete smooth manifold of codimension 1, therefore, $H_{0}=T_{u}S_{0}\bigoplus {\mathbb {R}}u$ for each $u\in T_{u}S_{0}$, where $T_{u}S_{0}=\{v\in H_{0}: \langle u, v\rangle _{0}=0\}$. Arguing as in Lemma 3.2, we have the following result.

Lemma 4.1

Let $V_{0}$ be given in (V1) and suppose that (f1)–(f4) are satisfied, then the following properties hold.

(a1):

For any $u\in H_{0}\backslash \{0\}$, let $g_{u}: {\mathbb {R}}^{+}\rightarrow {\mathbb {R}}$ be given by $g_{u}(t)=I_{0}(t u)$. Then there exists a unique $t_{u}>0$ such that $g'_{u}(t)>0$ in $(0, t_{u})$ and $g'_{u}(t)<0$ in $(t_{u}, \infty )$;

(a2):

There exists $\tau >0$ independent on u such that $t_{u}>\tau$ for all $u\in S_{0}$. Moreover, for each compact set ${\mathcal {W}}\subset S_{0}$ there is $C_{{\mathcal {W}}}$ such that $t_{u}\le C_{{\mathcal {W}}}$, for all $u\in {\mathcal {W}}$;

(a3):

The map ${\widehat{m}}: H_{0}\backslash \{0\}\rightarrow {\mathcal {N}}_{0}$ given by ${\widehat{m}}(u)=t_{u}u$ is continuous and $m_{0}={\widehat{m}}_{0}|_{S_{0}}$ is a homeomorphism between $S_{0}$ and ${\mathcal {N}}_{0}$. Moreover, $m^{-1}(u)=\frac{u}{\Vert u\Vert _{0}}\,$.

We shall consider the functional defined by

$$\begin{aligned} {\widehat{\Psi }}_{0}(u)=I_{0}({\widehat{m}}(u))\quad \text {and}\quad \Psi _{0}:={\widehat{\Psi }}_{0}|_{S_{0}}. \end{aligned}$$

Arguing as Proposition 9 and Corollary 10 in [35], we have that

Lemma 4.2

Let $V_{0}$ be given in (V1) and suppose that (f1)–(f4) are satisfied, then

(b1):

${\widehat{\Psi }}_{0}\in C^{1}(H_{0}\backslash \{0\}, {\mathbb {R}})$ and

$$\begin{aligned} {\widehat{\Psi }}_{0}'(u)v=\frac{\Vert {\widehat{m}}(u)\Vert _{0}}{\Vert u\Vert _{0}}I'_{0}({\widehat{m}}(u))[v],\,\,\,\forall \, u\in H_{0}\backslash \{0\}\,\,\text {and}\,\,\forall v\in H_{0}; \end{aligned}$$

(b2):

$\Psi _{0}\in C^{1}(S_{0}, {\mathbb {R}})$ and

$$\begin{aligned} \Psi _{0}'(u)v=\Vert m(u)\Vert _{0}I'_{0}({\widehat{m}}(u))[v],\,\,\,\forall v\in T_{u}S_{0}; \end{aligned}$$

(b3):

If $\{u_{n}\}$ is a $(PS)_{c}$ sequence of $\Psi _{0}$, then $\{m(u_{n})\}$ is a $(PS)_{c}$ sequence of $I_{0}$. If $\{u_{n}\}\subset {\mathcal {N}}_{0}$ is a bounded $(PS)_{c}$ sequence of $I_{0}$, then $\{m^{-1}(u_{n})\}$ is a $(PS)_{c}$ sequence of $\Psi _{0}$;

(b4):

u is a critical point of $\Psi _{0}$ if and only if m(u) is a critical point of $I_{0}$. Moreover, the corresponding critical values coincide and

$$\begin{aligned} \inf _{S_{0}}\Psi _{0}=\inf _{{\mathcal {N}}_{0}}I_{0}. \end{aligned}$$

Similar to the previous argument, we have the following variational characterization of the infimum of $I_{0}$ over ${\mathcal {N}}_{0}$:

$$\begin{aligned} c_{V_{0}}=\inf _{u\in {\mathcal {N}}_{0}}I_{0}(u)=\inf _{u\in H_{0}\backslash \{0\}}\sup _{t> 0}I_{0}(tu)=\inf _{u\in S_{0}}\sup _{t> 0}I_{0}(tu). \end{aligned}$$

The next result is useful in later arguments.

Lemma 4.3

Let $\{u_{n}\}\subset H_{0}$ be a $(PS)_{c}$ sequence for $I_{0}$ such that $u_{n}\rightharpoonup 0$. Then one of the following alternatives occurs:

• (i) $u_{n}\rightarrow 0$ in $H_{0}$ as $n\rightarrow +\infty$;

• (ii) there are a sequence $\{y_{n}\}\subset {\mathbb {R}}^{N}$ and constants R, $\beta >0$ such that

  $$\begin{aligned} \liminf _n \int _{B_{R}(y_{n})}\vert u_{n}\vert ^{2}dx\ge \beta . \end{aligned}$$

Proof

Assume that (ii) does not hold. Then, for every $R>0$, we have

$$\begin{aligned} \lim _n \sup _{y\in {\mathbb {R}}^{N}}\int _{B_{R}(y)}\vert u_{n}\vert ^{2}dx=0. \end{aligned}$$

Since $\{u_{n}\}$ is bounded in $H_{0}$, by Lions’ lemma [37], it follows that

$$\begin{aligned} u_{n}\rightarrow 0\,\,\text {in}\,\, L^{r}({\mathbb {R}}^{N}),\,\, 2< r<2^{*}. \end{aligned}$$

From the subcritical growth of f, we have

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}F(u_{n}^{2})dx=o_{n}(1)=\int _{{\mathbb {R}}^{N}}f(u_{n}^{2})u_{n}^{2}dx. \end{aligned}$$

Moreover, from $I'_{0}(u_{n})[u_{n}]\rightarrow 0$, it follows that

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}(\vert \nabla u_{n}\vert ^{2}+V_{0} u_{n}^{2})dx =\int _{{\mathbb {R}}^{N}}f(u_{n}^{2})u_{n}^{2}dx+o_{n}(1)=o_{n}(1). \end{aligned}$$

Thus, property (i) holds. $\square$

Remark 4.1

From Lemma 4.3 we see that if u is the weak limit of $(PS)_{c_{V_{0}}}$ sequence $\{u_{n}\}$ of the functional $I_{0}$, then we have $u\ne 0$. Otherwise we have that $u_{n}\rightharpoonup 0$ and if $u_{n}\not \rightarrow 0$, from Lemma 4.3 it follows that there are a sequence $\{y_{n}\}\subset {\mathbb {R}}^{N}$ and constants R, $\beta >0$ such that

$$\begin{aligned} \liminf _n \int _{B_{R}(y_{n})}\vert u_{n}\vert ^{2}dx\ge \beta >0. \end{aligned}$$

Then set $v_{n}(x)=u_{n}(x+z_{n})$, it is easy to see that $\{v_{n}\}$ is also a $(PS)_{c_{V_{0}}}$ sequence for the functional $I_{0}$, it is bounded, and there exists $v\in H_{0}$ such that $v_{n}\rightharpoonup v$ in $H_{0}$ with $v\ne 0$.

Lemma 4.4

Assume that V satisfies (V1), (V2) and f satisfies (f1)–(f4), then problem (4.1) has a positive ground state solution.

Proof

First of all, it is easy to show that $c_{V_{0}}>0$. Moreover, if $u_{0}\in {\mathcal {N}}_{0}$ satisfies $I_{0}(u_{0})=c_{V_{0}}$, then $m^{-1}(u_{0})\in S_{0}$ is a minimizer of $\Psi _{0}$, so that $u_{0}$ is a critical point of $I_{0}$ by Lemma 4.2. Now, we show that there exists a minimizer $u\in {\mathcal {N}}_{0}$ of $I_{0}|_{{\mathcal {N}}_{0}}$. Since $\inf _{S_{0}}\Psi _{0}=\inf _{{\mathcal {N}}_{0}}I_{0}=c_{V_{0}}$ and $S_{0}$ is a $C^{1}$ manifold, by Ekeland’s variational principle, there exists a sequence $\omega _{n}\subset S_{0}$ with $\Psi _{0}(\omega _{n})\rightarrow c_{V_{0}}$ and $\Psi '_{0}(\omega _{n})\rightarrow 0$ as $n\rightarrow \infty$. Put $u_{n}=m(\omega _{n})\in {\mathcal {N}}_{0}$ for $n\in N$. Then $I_{0}(u_{n})\rightarrow c_{V_{0}}$ and $I'_{0}(u_{n})\rightarrow 0$ as $n\rightarrow \infty$ by Lemma 4.2(b3). Similar to the proof of Lemma 3.4, it is easy to know that $\{u_{n}\}$ is bounded in $H_{0}$. Thus, we have $u_{n}\rightharpoonup u$ in $H_{0}$, $u_{n}\rightarrow u$ in $L_{\text {loc}}^{r}({\mathbb {R}}^{N}), 1\le r<2^{*}$ and $u_{n}\rightarrow u$ a.e. in ${\mathbb {R}}^{N}$, thus $I'_{0}(u)=0$. From Remark 4.1, we know that $u\ne 0$. Moreover,

$$\begin{aligned} c_{V_{0}}&\le I_{0}(u) =I_{0}(u)-\frac{1}{\theta }I'_{0}(u)[u]\\&= \Big (\frac{1}{2}-\frac{1}{\theta }\Big ) \Vert u\Vert _{0}^2 +\int _{{\mathbb {R}}^{N}}\Big (\frac{1}{\theta }f( u^{2})u^{2}-\frac{1}{2}F( u^{2})\Big )dx\\&\le \liminf _n\Big \{\Big (\frac{1}{2}-\frac{1}{\theta }\Big ) \Vert u_n\Vert _{0}^2 +\int _{{\mathbb {R}}^{N}}\Big (\frac{1}{\theta }f( u_{n}) u_{n}^{2}-\frac{1}{2}F( u_{n}^{2})\Big )dx\Big \}\\&= \liminf _n \Big \{ I_{0}(u_{n})-\frac{1}{\theta }I'_{0}(u_{n})[u_{n}]\Big \}\\&=c_{V_{0}}, \end{aligned}$$

thus, u is a ground state solution. From the assumption of f, $u\ge 0$, moreover, by [13, Proposition 6 and Proposition 7], we know that $u(x)>0$ for $x\in {\mathbb {R}}^{N}$. The proof is complete. $\square$

Note that, by [13, Proposition 3 and Proposition 4], the ground state solution of problem (4.1) is radially symmetric, which implies that every ground state solution decays exponentially at infinity with its gradient, and is $C^{2}({\mathbb {R}}^{N}, {\mathbb {R}})\cap L^\infty ({\mathbb {R}}^{N}, {\mathbb {R}})$.

Lemma 4.5

Let $(u_{n})\subset {\mathcal {N}}_{0}$ be such that $I_{0}(u_{n})\rightarrow c_{V_{0}}$. Then $(u_{n})$ has a convergent subsequence in $H_{0}$.

Proof

Since $(u_{n})\subset {\mathcal {N}}_{0}$, from Lemma 4.1(a3), Lemma 4.2(b4) and the definition of $c_{V_{0}}$, we have

$$\begin{aligned} v_{n}=m^{-1}(u_{n})= \frac{u_{n}}{\Vert u_{n}\Vert _{0}}\in S_{0}, \,\,\,\forall n\in N, \end{aligned}$$

and

$$\begin{aligned} \Psi _{0}(v_{n})=I_{0}(u_{n})\rightarrow c_{V_{0}}=\inf _{u\in S_{0}}\Psi _{0}(u). \end{aligned}$$

Since $S_{0}$ is a complete $C^{1}$ manifold, by Ekeland’s variational principle, there exists a sequence $\{{\tilde{v}}_{n}\}\subset S_{0}$ such that $\{{\tilde{v}}_{n}\}$ is a $(PS)_{c_{V_{0}}}$ sequence for $\Psi _{0}$ on $S_{0}$ and

$$\begin{aligned} \Vert {\tilde{v}}_{n}-v_{n}\Vert _{0}=o_{n}(1). \end{aligned}$$

Similar to the proof of Lemma 4.4, we may obtain the conclusion of this lemma. $\square$

4.2 The technical results

In this subsection, we prove a multiplicity result for the modified problem (3.2) using the Ljusternik–Schnirelmann category theory. In order to get it, we first provide some useful preliminaries.

Let $\delta >0$ be such that $M_{\delta }\subset \Lambda$, $\omega \in H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$ be a positive ground state solution of the limit problem (4.1), and $\eta \in C^{\infty }({\mathbb {R}}^{+}, [0, 1])$ be a nonincreasing cut-off function defined in $[0, +\infty )$ such that $\eta (t)=1$ if $0\le t\le \delta /{2}$ and $\eta (t)=0$ if $t\ge \delta$.

For any $y\in M$, let us introduce the function

$$\begin{aligned} \Psi _{\varepsilon , y}(x):=\eta (\vert \varepsilon x-y\vert )\omega \Big (\frac{\varepsilon x-y}{\varepsilon }\Big )\exp \Big (i\tau _{y}\Big (\frac{\varepsilon x-y}{\varepsilon }\Big )\Big ), \end{aligned}$$

where

$$\begin{aligned} \tau _{y}(x):=\sum _{i}^{N}A_{i}(y)x_{i}. \end{aligned}$$

Let $t_{\varepsilon }>0$ be the unique positive number such that

$$\begin{aligned} \max _{t\ge 0}J_{\varepsilon }(t\Psi _{\varepsilon , y})=J_{\varepsilon }(t_{\varepsilon }\Psi _{\varepsilon , y}). \end{aligned}$$

Note that $t_{\varepsilon }\Psi _{\varepsilon , y}\in {\mathcal {N}}_{\varepsilon }$.

Let us define $\Phi _{\varepsilon }:M\rightarrow {\mathcal {N}}_{\varepsilon }$ as

$$\begin{aligned} \Phi _{\varepsilon }(y):=t_{\varepsilon }\Psi _{\varepsilon , y}. \end{aligned}$$

By construction, $\Phi _{\varepsilon }(y)$ has compact support for any $y\in M$. Moreover, the energy of the above functions has the following behavior as $\varepsilon \rightarrow 0^{+}$.

Lemma 4.6

The limit

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^{+}}J_{\varepsilon }(\Phi _{\varepsilon }(y))=c_{V_{0}} \end{aligned}$$

holds uniformly in $y\in M$.

Proof

Assume by contradiction that the statement is false. Then there exist $\delta _{0}>0$, $(y_{n})\subset M$ and $\varepsilon _{n}\rightarrow 0^+$ satisfying

$$\begin{aligned} \Big \vert J_{\varepsilon _{n}}(\Phi _{\varepsilon _{n}}(y_{n}))-c_{V_{0}}\Big \vert \ge \delta _{0}. \end{aligned}$$

For simplicity, we write $\Phi _{n}$, $\Psi _{n}$ and $t_{n}$ for $\Phi _{\varepsilon _{n}}(y_{n})$, $\Psi _{\varepsilon _{n}, y_{n}}$ and $t_{\varepsilon _{n}}$, respectively.

We can check that

$$\begin{aligned} \Vert \Psi _{n}\Vert _{\varepsilon _{n}}^{2}\rightarrow \int _{{\mathbb {R}}^{N}}(\vert \nabla \omega \vert ^{2}+V_{0} \omega ^{2})dx \text { as } n\rightarrow +\infty . \end{aligned}$$

(4.2)

Indeed, by a change of variable of $z=(\varepsilon _{n} x-y_{n})/\varepsilon _{n}$, the Lebesgue Dominated Convergence Theorem, the continuity of V and $y_{n}\in M\subset \Lambda$(which is bounded), we deduce that

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}V(\varepsilon _{n} x)\vert \Psi _{n}\vert ^{2}dx= \int _{{\mathbb {R}}^{N}}V(\varepsilon _{n} z+y_{n})\vert \eta (\vert \varepsilon _{n} z \vert )\omega (z)\vert ^{2}dx \rightarrow V_{0} \int _{{\mathbb {R}}^{N}}\omega ^{2}dx \text { as } n\rightarrow +\infty . \end{aligned}$$

Moreover, by the same change of variable $z=(\varepsilon _{n}x-y_n)/\varepsilon _{n}$, we also have

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}\vert \nabla _{A_{\varepsilon _{n}}}\Psi _{n}\vert ^{2}dx&= \varepsilon _{n}^2 \int _{{\mathbb {R}}^{N}}| \eta '(\vert \varepsilon _{n} z\vert )\omega (z)|^{2} dz +\int _{{\mathbb {R}}^{N}} | \eta (\vert \varepsilon _{n} z\vert )\nabla \omega (z)|^{2}dz\\&\quad +\int _{{\mathbb {R}}^{N}}\Big | \eta (\vert \varepsilon _{n} z\vert )\Big (A(y_n)-A(\varepsilon _{n}z+y_n)\Big )\omega (z)\Big |^{2}dz\\&\quad +2\varepsilon _{n} \int _{{\mathbb {R}}^{N}} \eta (\vert \varepsilon _{n} z\vert ) \eta '(\vert \varepsilon _{n} z\vert )\omega (z)\nabla \omega (z)\cdot \frac{z}{|z|}dz. \end{aligned}$$

It is clear that

$$\begin{aligned} \lim _n \int _{{\mathbb {R}}^{N}}| \eta (\vert \varepsilon _{n} z\vert )\nabla \omega (z)|^{2}dz =\int _{{\mathbb {R}}^{N}}\vert \nabla \omega (z)\vert ^{2}dz. \end{aligned}$$

Moreover, using the definition of $\eta$, the Hölder continuity with exponent $\alpha \in (0, 1]$ of A, the exponential decay of $\omega$, and the Lebesgue Dominated Convergence Theorem, we can infer

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}|\eta '(\vert \varepsilon _{n} z\vert )\omega (z)|^{2}dz&=o_n(1), \\ \int _{{\mathbb {R}}^{N}}|\eta (\vert \varepsilon _{n} z\vert )\eta '(\vert \varepsilon _{n} z\vert )\omega (z)\nabla \omega (z)|dz&=o_n(1), \end{aligned}$$

and

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}\Big | \eta (\vert \varepsilon _{n} z\vert )\Big (A(y_n)-A(\varepsilon _{n}z+y_n)\Big )\omega (z)\Big |^{2}dz\le C\varepsilon _{n}^{2\alpha }\int _{\vert \varepsilon _{n} z\vert \le \delta } \omega ^{2}(z)\vert z\vert ^{2\alpha }dz=o_{n}(1), \end{aligned}$$

obtaining (4.2).

On the other hand, since $J'_{\varepsilon _{n}}(t_{n}\Psi _{n})(t_{n}\Psi _{n})=0$, by the change of variables $z=(\varepsilon _{n} x-y_{n})/\varepsilon _{n}$, observe that, if $z\in B_{\delta /\varepsilon _{n}}(0)$, then $\varepsilon _{n} z+y_{n}\in B_{\delta }(y_{n})\subset M_{\delta }\subset \Lambda$, we have

$$\begin{aligned} \Vert \Psi _{n}\Vert _{\varepsilon _{n}}^{2}&= \int _{{\mathbb {R}}^{N}} g(\varepsilon _{n}z+y_{n}, t_{n}^2\eta ^2(\vert \varepsilon _{n} z\vert )\omega ^2(z)) \eta ^2(\vert \varepsilon _{n} z\vert )\omega ^2(z) dz\\&= \int _{{\mathbb {R}}^{N}}f(t_{n}^2\eta ^2(\vert \varepsilon _{n} z\vert )\omega ^2(z)) \eta ^2(\vert \varepsilon _{n} z\vert )\omega ^2(z) dz\\&\ge \int _{B_{\delta /(2\varepsilon _{n})}(0)}f( t_{n}^2 \omega ^2(z))\omega ^{2}(z)dz \\&\ge \int _{B_{\delta /2}(0)}f( t_{n}^2\omega ^2(z))\omega ^{2}(z)dz\\&\ge f( t_{n}^2\gamma ^{2})\int _{B_{\delta /2}(0)}\omega ^{2}(z)dz \end{aligned}$$

for all n large enough and where $\gamma =\min \{\omega (z): \vert z\vert \le \delta /2\}$.

If $t_{n}\rightarrow +\infty$, by (f4) we deduce that $\Vert \Psi _{n}\Vert _{\varepsilon _{n}}^{2}\rightarrow +\infty$ which contradicts (4.2). Therefore, up to a subsequence, we may assume that $t_{n}\rightarrow t_{0}\ge 0$.

If $t_{n}\rightarrow 0$, using the fact that f is increasing and the Lebesgue Dominated Convergence Theorem, we obtain that

$$\begin{aligned} \Vert \Psi _{n}\Vert _{\varepsilon _{n}}^{2}=\int _{{\mathbb {R}}^{N}}f(t_{n}^2 \eta ^2(\vert \varepsilon _{n} z\vert )\omega ^2(z)) \eta ^2(\vert \varepsilon _{n} z\vert )\omega ^2(z)dz\rightarrow 0, \text { as } n\rightarrow +\infty , \end{aligned}$$

which contradicts (4.2). Thus, we have $t_{0}>0$ and

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}(\vert \nabla \omega \vert ^{2}+V_{0}\omega ^2)dx=\int _{{\mathbb {R}}^{N}}f( t_{0}\omega ^{2})\omega ^{2}dx, \end{aligned}$$

so that $t_{0}\omega \in {\mathcal {N}}_{V_{0}}$. Since $\omega \in {\mathcal {N}}_{V_{0}}$, we obtain that $t_{0}=1$ and so, using the Lebesgue Dominated Convergence Theorem, we get

$$\begin{aligned} \lim _{n}\int _{{\mathbb {R}}^{N}}F(\vert t_{n}\Psi _{n}\vert ^{2})dx=\int _{{\mathbb {R}}^{N}}F( \omega ^{2})dx. \end{aligned}$$

Hence

$$\begin{aligned} \lim _{n}J_{\varepsilon _{n}}(\Phi _{\varepsilon _{n}}(y_{n}))=I_{0}(\omega )=c_{V_{0}} \end{aligned}$$

which is a contradiction and the proof is complete. $\square$

Now we define the barycenter map.

Let $\rho >0$ be such that $M_{\delta }\subset B_{\rho }$ and consider $\Upsilon : {\mathbb {R}}^{N}\rightarrow {\mathbb {R}}^{N}$ defined by setting

$$\begin{aligned} \Upsilon (x):=\left\{ \begin{array}{ll} x,&{}\quad \text {if}\,\,\vert x\vert <\rho ,\\ \rho x/\vert x\vert ,&{}\quad \text {if}\,\,\vert x\vert \ge \rho . \end{array} \right. \end{aligned}$$

The barycenter map $\beta _{\varepsilon }: {\mathcal {N}}_{\varepsilon }\rightarrow {\mathbb {R}}^{N}$ is defined by

$$\begin{aligned} \beta _{\varepsilon }(u):=\frac{1}{\Vert u\Vert _2^{2}}\int _{{\mathbb {R}}^{N}}\Upsilon (\varepsilon x)\vert u(x)\vert ^{2} dx. \end{aligned}$$

We have the following lemma.

Lemma 4.7

The limit

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^{+}}\beta _{\varepsilon }(\Phi _{\varepsilon }(y))=y \end{aligned}$$

holds uniformly in $y\in M$.

Proof

Assume by contradiction that there exist $\kappa >0$, $(y_{n})\subset M$ and $\varepsilon _{n}\rightarrow 0$ such that

$$\begin{aligned} \vert \beta _{\varepsilon _{n}}(\Phi _{\varepsilon _{n}}(y_{n}))-y_{n}\vert \ge \kappa . \end{aligned}$$

(4.3)

Using the change of variable $z=(\varepsilon _{n} x-y_{n})/\varepsilon _{n}$, we can see that

$$\begin{aligned} \beta _{\varepsilon _{n}}(\Phi _{\varepsilon _{n}}(y_{n}))=y_{n}+\frac{\displaystyle \int _{{\mathbb {R}}^{N}}(\Upsilon (\varepsilon _{n} z+y_{n})-y_{n}) \eta ^2(\vert \varepsilon _{n} z\vert ) \omega ^2(z) dz}{\displaystyle \int _{{\mathbb {R}}^{N}}\eta ^2(\vert \varepsilon _{n} z\vert ) \omega ^2(z) dz}. \end{aligned}$$

Taking into account $(y_{n})\subset M\subset M_{\delta }\subset B_{\rho }$ and the Lebesgue Dominated Convergence Theorem, we can obtain that

$$\begin{aligned} \vert \beta _{\varepsilon _{n}}(\Phi _{\varepsilon _{n}}(y_{n}))-y_{n}\vert =o_{n}(1), \end{aligned}$$

which contradicts (4.3). $\square$

Now, we prove the following useful compactness result.

Proposition 4.1

Let $\varepsilon _{n}\rightarrow 0^+$ and $(u_{n})\subset {\mathcal {N}}_{\varepsilon _{n}}$ be such that $J_{\varepsilon _{n}}(u_{n})\rightarrow c_{V_{0}}$. Then there exists $({\tilde{y}}_{n})\subset {\mathbb {R}}^{N}$ such that the sequence $(\vert v_n\vert )\subset H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, where $v_{n}(x):= u_{n}(x+{\tilde{y}}_{n})$, has a convergent subsequence in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$. Moreover, up to a subsequence, $y_{n}:=\varepsilon _{n}{\tilde{y}}_{n}\rightarrow y\in M$ as $n\rightarrow +\infty$.

Proof

The proof of this proposition can be found in [3]. However, for the reader’s convenience, we give in what follows the details of the proof. Since $J'_{\varepsilon _{n}}(u_{n}) [u_{n}]=0$ and $J_{\varepsilon _{n}}(u_{n})\rightarrow c_{V_{0}}$, arguing as in the proof of Lemma 3.4, we can prove that there exists $C>0$ such that $\Vert u_{n}\Vert _{\varepsilon _{n}}\le C$ for all $n\in {\mathbb {N}}$.

Arguing as in the proof of Lemma 3.2 and recalling that $c_{V_{0}}>0$, we have that there exist a sequence $\{{\tilde{y}}_{n}\}\subset {\mathbb {R}}^{N}$ and constants R, $\beta >0$ such that

$$\begin{aligned} {\liminf _n}\int _{B_{R}({\tilde{y}}_{n})}\vert u_{n}\vert ^{2}dx\ge \beta . \end{aligned}$$

(4.4)

Now, let us consider the sequence $\{\vert v_n\vert \}\subset H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, where $v_{n}(x):= u_{n}(x+{\tilde{y}}_{n})$. By the diamagnetic inequality (2.1), we get that $\{\vert v_{n}\vert \}$ is bounded in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, and using (4.4), we may assume that $\vert v_{n}\vert \rightharpoonup v$ in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$ for some $v\ne 0$.

Let $t_{n}>0$ be such that ${\tilde{v}}_{n}:=t_{n}\vert v_{n}\vert \in {\mathcal {N}}_{V_{0}}$, and set $y_{n}:=\varepsilon _{n}{\tilde{y}}_{n}$.

By the diamagnetic inequality (2.1), we have

$$\begin{aligned} c_{V_{0}}\le I_{0}({\tilde{v}}_{n})\le \max _{t\ge 0}J_{\varepsilon _{n}}(tu_{n})=J_{\varepsilon _{n}}(u_{n})=c_{V_{0}}+o_{n}(1), \end{aligned}$$

which yields $I_{0}({\tilde{v}}_{n})\rightarrow c_{V_{0}}$ as $n\rightarrow +\infty$.

Since the sequences $\{\vert v_{n}\vert \}$ and $\{{\tilde{v}}_{n}\}$ are bounded in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$ and $\vert v_{n}\vert \not \rightarrow 0$ in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, then $(t_{n})$ is also bounded and so, up to a subsequence, we may assume that $t_{n}\rightarrow t_{0}\ge 0$.

We claim that $t_{0}>0$. Indeed, if $t_0=0$, then, since $(|v_{n}|)$ is bounded, we have ${\tilde{v}}_{n}\rightarrow 0$ in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, that is $I_{0}({\tilde{v}}_{n})\rightarrow 0$, which contradicts $c_{V_{0}}>0$. Thus, up to a subsequence, we may assume that ${\tilde{v}}_{n}\rightharpoonup {\tilde{v}}:=t_{0} v\ne 0$ in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, and, by Lemma 4.5, we can deduce that ${\tilde{v}}_{n}\rightarrow {\tilde{v}}$ in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, which gives $\vert v_{n}\vert \rightarrow v$ in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$.

Now we show the final part, namely that $\{y_{n}\}$ has a subsequence such that $y_{n}\rightarrow y\in M$. Assume by contradiction that $\{y_{n}\}$ is not bounded and so, up to a subsequence, $\vert y_{n}\vert \rightarrow +\infty$ as $n\rightarrow +\infty$. Choose $R>0$ such that $\Lambda \subset B_{R}(0)$. Then for n large enough, we have $\vert y_{n}\vert >2R$, and, for any $x\in B_{R/\varepsilon _{n}}(0)$,

$$\begin{aligned} \vert \varepsilon _{n} x+y_{n}\vert \ge \vert y_{n}\vert -\varepsilon _{n}\vert x\vert >R. \end{aligned}$$

Since $u_{n}\in {\mathcal {N}}_{\varepsilon _{n}}$, using (V1) and the diamagnetic inequality (2.1), we get that

$$\begin{aligned} \begin{aligned} \int _{{\mathbb {R}}^{N}}(\vert \nabla \vert v_{n}\vert \vert ^{2}+ V_{0}\vert v_{n}\vert ^{2})dx&\le \int _{{\mathbb {R}}^{N}}g(\varepsilon _{n} x+y_{n}, \vert v_{n}\vert ^{2})\vert v_{n}\vert ^{2}dx\\&\le \int _{B_{R/\varepsilon _{n}}(0)}{\tilde{f}}(\vert v_{n}\vert ^{2})\vert v_{n}\vert ^{2}dx+\int _{B^{c}_{R/\varepsilon _{n}}(0)}f(\vert v_{n}\vert ^{2})\vert v_{n}\vert ^{2}dx. \end{aligned} \end{aligned}$$

(4.5)

Since $\vert v_{n}\vert \rightarrow v$ in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$ and ${\tilde{f}}(t)\le V_{0}/{K}$, we can see that (4.5) yields

$$\begin{aligned} \min \Big \{1, V_{0}\Big (1-\frac{1}{K}\Big )\Big \}\int _{{\mathbb {R}}^{N}}(\vert \nabla \vert v_{n}\vert \vert ^{2}+ \vert v_{n}\vert ^{2})dx=o_{n}(1), \end{aligned}$$

that is $\vert v_{n}\vert \rightarrow 0$ in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, which contradicts to $v\not \equiv 0$.

Therefore, we may assume that $y_{n}\rightarrow y_{0}\in {\mathbb {R}}^{N}$. Assume by contradiction that $y_{0}\not \in {\overline{\Lambda }}$. Then there exists $r>0$ such that for every n large enough we have that $|y_n -y_0|<r$ and $B_{2r}(y_0)\subset {\overline{\Lambda }}^c$. Then, if $x\in B_{r/\varepsilon _n}(0)$, we have that $|\varepsilon _n x + y_n - y_0| <2r$ so that $\varepsilon _n x + y_n\in {\overline{\Lambda }}^c$ and so, arguing as before, we reach a contradiction. Thus, $y_{0}\in {\overline{\Lambda }}$.

To prove that $V(y_{0})=V_{0}$, we suppose by contradiction that $V(y_{0})>V_{0}$. Using the Fatou’s lemma, the change of variable $z=x+{\tilde{y}}_n$ and $\max _{t\ge 0}J_{\varepsilon _{n}}(t u_{n})=J_{\varepsilon _{n}}(u_{n})$, we obtain

$$\begin{aligned} c_{V_{0}}=I_{0}({\tilde{v}})&<\frac{1}{2}\int _{{\mathbb {R}}^{N}}(\vert \nabla {\tilde{v}}\vert ^{2}+ V(y_{0})\vert {\tilde{v}}\vert ^{2})dx-\frac{1}{2}\int _{{\mathbb {R}}^{N}}F(\vert {\tilde{v}}\vert ^{2})dx\\&\le {\liminf _n}\Big (\frac{1}{2}\int _{{\mathbb {R}}^{N}}(\vert \nabla {\tilde{v}}_{n}\vert ^{2}+ V(\varepsilon _{n}x+y_{n})\vert {\tilde{v}}_{n}\vert ^{2})dx-\frac{1}{2}\int _{{\mathbb {R}}^{N}}F(\vert {\tilde{v}}_{n}\vert ^{2})dx\Big )\\&={\liminf _n}\Big (\frac{t^{2}_{n}}{2}\int _{{\mathbb {R}}^{N}}(\vert \nabla \vert u_{n}\vert \vert ^{2}+ V(\varepsilon _{n}z)\vert u_{n}\vert ^{2})dz-\frac{1}{2}\int _{{\mathbb {R}}^{N}}F(\vert t_{n}u_{n}\vert ^{2})dz\Big )\\&\le {\liminf _n}J_{\varepsilon _{n}}(t_{n}u_{n})\le {\liminf _n}J_{\varepsilon _{n}}(u_{n})= c_{V_{0}} \end{aligned}$$

which is impossible and the proof is complete. $\square$

Let now

$$\begin{aligned} \tilde{{\mathcal {N}}}_{\varepsilon }:=\{u\in {\mathcal {N}}_{\varepsilon }: J_{\varepsilon }(u)\le c_{V_{0}}+h(\varepsilon )\}, \end{aligned}$$

where $h: {\mathbb {R}}^{+}\rightarrow {\mathbb {R}}^{+}$, $h(\varepsilon )\rightarrow 0$ as $\varepsilon \rightarrow 0^+$.

Fixed $y\in M$, since, by Lemma 4.6, $\vert J_{\varepsilon }(\Phi _{\varepsilon }(y))-c_{V_{0}}\vert \rightarrow 0$ as $\varepsilon \rightarrow 0^+$, we get that $\tilde{{\mathcal {N}}}_{\varepsilon }\ne \emptyset$ for any $\varepsilon >0$ small enough.

We have the following relation between $\tilde{{\mathcal {N}}}_{\varepsilon }$ and the barycenter map.

Lemma 4.8

We have

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0^{+}}\sup _{u\in \tilde{{\mathcal {N}}}_{\varepsilon }}{\text {dist}}(\beta _{\varepsilon }(u), M_{\delta })=0. \end{aligned}$$

Proof

Let $\varepsilon _{n}\rightarrow 0^+$ as $n\rightarrow +\infty$. For any $n\in {\mathbb {N}}$, there exists $u_{n}\in \tilde{{\mathcal {N}}}_{\varepsilon _n}$ such that

$$\begin{aligned} \sup _{u\in \tilde{{\mathcal {N}}}_{\varepsilon _n}}\inf _{y\in M_{\delta }}|\beta _{\varepsilon _n}(u)- y|=\inf _{y\in M_{\delta }}|\beta _{\varepsilon _n}(u_{n})- y|+o_{n}(1). \end{aligned}$$

Therefore, it is enough to prove that there exists $(y_{n})\subset M_{\delta }$ such that

$$\begin{aligned} \lim _{n}\vert \beta _{\varepsilon _n}(u_{n})-y_{n}\vert =0. \end{aligned}$$

By the diamagnetic inequality (2.1), we can see that $I_{0}(t\vert u_{n}\vert )\le J_{\varepsilon _{n}}(t u_{n})$ for any $t\ge 0$. Therefore, recalling that $\{u_{n}\}\subset \tilde{{\mathcal {N}}}_{\varepsilon _n}\subset {\mathcal {N}}_{\varepsilon _n}$, we can deduce that

$$\begin{aligned} c_{V_{0}}\le \max _{t\ge 0}I_{0}(t\vert u_{n}\vert )\le \max _{t\ge 0}J_{\varepsilon _{n}}(t u_{n})=J_{\varepsilon _{n}}(u_{n})\le c_{V_{0}}+h(\varepsilon _{n}) \end{aligned}$$

(4.6)

which implies that $J_{\varepsilon _{n}}(u_{n})\rightarrow c_{V_{0}}$ as $n\rightarrow +\infty$. Then Proposition 4.1 implies that there exists $\{{\tilde{y}}_{n}\}\subset {\mathbb {R}}^{N}$ such that $y_{n}=\varepsilon _{n}{\tilde{y}}_{n}\in M_{\delta }$ for n large enough. Thus, making the change of variable $z=x-{\tilde{y}}_n$, we get

$$\begin{aligned} \beta _{\varepsilon _{n}}(u_{n})=y_{n}+\frac{\int _{{\mathbb {R}}^{N}}(\Upsilon (\varepsilon _{n} z+y_{n})-y_{n})\vert u_{n}(z+{\tilde{y}}_{n})\vert ^{2} dz}{\int _{{\mathbb {R}}^{N}}\vert u_{n}(z+{\tilde{y}}_{n})\vert ^{2} dz}. \end{aligned}$$

Since, up to a subsequence, $\vert u_{n}\vert (\cdot +{\tilde{y}}_{n})$ converges strongly in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$ and $\varepsilon _{n}z+y_{n}\rightarrow y\in M$ for any $z\in {\mathbb {R}}^{N}$, we conclude the proof. $\square$

4.3 Multiplicity of solutions for problem (3.2)

Finally, we present a relation between the topology of M and the number of solutions of the modified problem (3.2).

Theorem 4.1

For any $\delta >0$ such that $M_{\delta }\subset \Lambda$, there exists ${\tilde{\varepsilon }}_{\delta }>0$ such that, for any $\varepsilon \in (0, {\tilde{\varepsilon }}_{\delta })$, problem (3.2) has at least $\text {cat}_{M_{\delta }}(M)$ nontrivial solutions.

Proof

For any $\epsilon >0$, we define the function $\pi _{\epsilon }: M\rightarrow S_{\varepsilon }^{+}$ by

$$\begin{aligned} \pi _{\epsilon }(y)=m_{\varepsilon }^{-1}(\Phi _{\epsilon }(y)),\,\, \forall y\in M. \end{aligned}$$

By Lemma 4.6 and Lemma 3.3(B4), we obtain

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \Psi _{\epsilon }(\pi _{\epsilon }(y))=\lim _{\epsilon \rightarrow 0} J_{\epsilon }(\Phi _{\epsilon }(y))=c_{V_{0}},\,\,\text {uniformly in}\,\,y\in M. \end{aligned}$$

Hence, there is a number ${\hat{\epsilon }}>0$ such that the set ${\tilde{S}}_{\varepsilon }^{+}:=\{u\in S_{\varepsilon }^{+}: \Psi _{\varepsilon }(u)\le c_{V_{0}}+h(\varepsilon )\}$ is nonempty, for all $\epsilon \in (0, {\hat{\epsilon }})$, since $\pi _{\epsilon }(M)\subset {\tilde{S}}_{\varepsilon }^{+}$. Here h is given in the definition of $\tilde{{\mathcal {N}}}_{\varepsilon }$.

Given $\delta >0$, by Lemma 4.6, Lemma 3.2(A3), Lemma 4.7, and Lemma 4.8, we can find ${\tilde{\varepsilon }}_{\delta }>0$ such that for any $\varepsilon \in (0, {\tilde{\varepsilon }}_{\delta })$, the following diagram

$$\begin{aligned} M \xrightarrow {\Phi _{\varepsilon }}\Phi _{\varepsilon }(M)\xrightarrow {m_{\varepsilon }^{-1}}\pi _{\epsilon }(M)\xrightarrow {m_{\varepsilon }} \Phi _{\varepsilon }(M)\xrightarrow {\beta _{\varepsilon }} M_{\delta } \end{aligned}$$

is well defined and continuous. From Lemma 4.7, we can choose a function $\Theta (\epsilon , z)$ with $\vert \Theta (\epsilon , z)\vert <\frac{\delta }{2}$ uniformly in $z\in M$, for all $\epsilon \in (0, {\hat{\epsilon }})$ such that $\beta _{\varepsilon }(\Phi _{\varepsilon }(z))=z+\Theta (\epsilon , z)$ for all $z\in M$. Define $H(t, z)=z+(1-t)\Theta (\epsilon , z)$. Then $H: [0, 1]\times M \rightarrow M_{\delta }$ is continuous. Clearly, $H(0, z)=\beta _{\varepsilon }(\Phi _{\varepsilon }(z))$, $H(1, z)=z$ for all $z\in M$. That is, H(t, z) is a homotopy between $\beta _{\varepsilon }\circ \Phi _{\varepsilon }=(\beta _{\varepsilon }\circ m_{\varepsilon })\circ \pi _{\epsilon }$ and the embedding $\iota : M\rightarrow M_{\delta }$. Thus, this fact implies that

$$\begin{aligned} \text {cat}_{\pi _{\epsilon }(M)}(\pi _{\epsilon }(M))\ge \text {cat}_{M_{\delta }}(M). \end{aligned}$$

(4.7)

By Corollary 3.1 and the abstract category theorem [35], $\Psi _{\varepsilon }$ has at least $\text {cat}_{\pi _{\epsilon }(M)}(\pi _{\epsilon }(M))$ critical points on $S_{\varepsilon }^{+}$. Therefore, from Lemma 3.3(B4) and (4.7), we have that $J_{\varepsilon }$ has at least $\text {cat}_{M_{\delta }}(M)$ critical points in $\tilde{{\mathcal {N}}}_{\varepsilon }$ which implies that problem (3.2) has at least $\text {cat}_{M_{\delta }}(M)$ solutions. $\square$

5 Proof of Theorem 1.1

In this section we prove our main result. The idea is to show that the solutions $u_\varepsilon$ obtained in Theorem 4.1 satisfy

$$\begin{aligned} \vert u_{\varepsilon }(x)\vert ^2\le a \text { for } x\in \Lambda ^{c}_{\varepsilon } \end{aligned}$$

for $\varepsilon$ small. The key ingredient is the following result.

Lemma 5.1

Let $\varepsilon _{n}\rightarrow 0^{+}$ and $u_{n}\in \tilde{{\mathcal {N}}}_{\varepsilon _{n}}$ be a solution of problem (3.2) for $\varepsilon =\varepsilon _n$. Then $J_{\varepsilon _{n}}(u_{n})\rightarrow c_{V_{0}}$. Moreover, there exists $\{{\tilde{y}}_{n}\}\subset {\mathbb {R}}^N$ such that, if $v_{n}(x):= u_{n}(x+{\tilde{y}}_{n})$, we have that $\{\vert v_{n}\vert \}$ is bounded in $L^{\infty }({\mathbb {R}}^{N}, {\mathbb {R}})$ and

$$\begin{aligned} \lim _{\vert x\vert \rightarrow +\infty } \vert v_{n}(x)\vert =0\quad \text {uniformly in } n\in {\mathbb {N}}. \end{aligned}$$

Proof

The proof of this lemma can be found in [3], for the convenience of the readers, we give the proof here. Since $J_{\varepsilon _{n}}(u_{n})\le c_{V_{0}}+h(\varepsilon _{n})$ with $\lim _{n}h(\varepsilon _{n})=0$, we can argue as in the proof of Lemma 4.8 (see (4.6)) to conclude that $J_{\varepsilon _{n}}(u_{n}) \rightarrow c_{V_{0}}$. Thus, by Proposition 4.1, we obtain the existence of a sequence $\{{\tilde{y}}_{n}\}\subset {\mathbb {R}}^{N}$ such that $\{\vert v_n\vert \}\subset H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$, where $v_{n}(x):=u_{n}(x+{\tilde{y}}_{n})$, has a convergent subsequence in $H^{1}({\mathbb {R}}^{N}, {\mathbb {R}})$. Moreover, up to a subsequence, $y_{n}:=\varepsilon _{n}{\tilde{y}}_{n}\rightarrow y\in M$ as $n\rightarrow +\infty$.

For any $R>0$ and $0<r\le R/2$, let $\eta \in C^{\infty }({\mathbb {R}}^{N})$, $0\le \eta \le 1$ with $\eta (x)=1$ if $\vert x\vert \ge R$ and $\eta (x)=0$ if $\vert x\vert \le R-r$ and $\vert \nabla \eta \vert \le 2/r$.

For each $n\in {\mathbb {N}}$ and $L>0$, we consider the functions

$$\begin{aligned} v_{L, n}(x):= {\left\{ \begin{array}{ll} \vert v_{n}(x)\vert &{} \text {if }\vert v_{n}(x)\vert \le L,\\ L &{} \text {if }\vert v_{n}(x)\vert > L, \end{array}\right. } \qquad z_{L, n}:=\eta ^{2}v_{L, n}^{2(\beta -1)}v_{n}, \quad \text {and}\quad w_{L, n}:=\eta v_{L, n}^{\beta -1}\vert v_{n}\vert , \end{aligned}$$

where $\beta >1$ will be determined later.

Since, by the diamagnetic inequality (2.1) we have that

$$\begin{aligned} {\text {Re}}(\nabla _{A_{\varepsilon _{n}}(\cdot + {\tilde{y}}_n)}v_{n} \cdot \overline{\nabla _{A_{\varepsilon _{n}}(\cdot + {\tilde{y}}_n)}z_{L, n}})&=\eta ^{2}v_{L, n}^{2(\beta -1)} \vert \nabla _{A_{\varepsilon _{n}}(\cdot + {\tilde{y}}_n)}v_{n}\vert ^{2} + {\text {Re}}(\nabla v_{n} \overline{v_{n}}) \nabla \left( \eta ^2 v_{L, n}^{2(\beta -1)}\right) \\&= \eta ^{2}v_{L, n}^{2(\beta -1)} \vert \nabla _{A_{\varepsilon _{n}}(\cdot + {\tilde{y}}_n)}v_{n}\vert ^{2} + \vert v_{n}\vert \nabla \vert v_{n}\vert \nabla \left( \eta ^2 v_{L, n}^{2(\beta -1)}\right) \\&\ge \eta ^{2}v_{L, n}^{2(\beta -1)} \vert \nabla \vert v_{n}\vert \vert ^{2}+2\eta \nabla \eta v_{L, n}^{2(\beta -1)}\vert v_{n}\vert \nabla \vert v_n\vert , \end{aligned}$$

taking $z_{L, n}$ as a test function, we have

$$\begin{aligned} \begin{aligned}&\int _{{\mathbb {R}}^{N}}\vert \nabla \vert v_{n}\vert \vert ^{2}\eta ^{2}v_{L, n}^{2(\beta -1)}dx +2 \int _{{\mathbb {R}}^{N}} \eta \nabla \eta v_{L, n}^{2(\beta -1)}\vert v_{n}\vert \nabla \vert v_n\vert dx \\&\qquad + \int _{{\mathbb {R}}^{N}}V(\varepsilon _{n} x+\varepsilon _{n}{\tilde{y}}_{n})\eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2}dx \le {\text {Re}}\int _{{\mathbb {R}}^{N}}(\nabla _{A_{\varepsilon _{n}}(\cdot + {\tilde{y}}_n)} v_{n} \cdot \overline{\nabla _{A_{\varepsilon _{n}}(\cdot + {\tilde{y}}_n)} z_{L, n}}) dx\\&\qquad +{\text {Re}}\int _{{\mathbb {R}}^{N}}V(\varepsilon _{n} x+\varepsilon _{n}{\tilde{y}}_{n})v_{n}\overline{z_{L, n}}dx\\&\quad = \int _{{\mathbb {R}}^{N}}g({\varepsilon _{n} x+\varepsilon _{n}{\tilde{y}}_{n}}, \vert v_{n}\vert ^{2})\eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} dx. \end{aligned} \end{aligned}$$

(5.1)

We first deal with the case $N\ge 3$. From the definition of g, for any $0<\zeta <V_{0}$ small, there exists $C_{\zeta }>0$ such that

$$\begin{aligned} g(x, t^{2})t^{2}\le \zeta t^{2}+C_{\zeta }\vert t\vert ^{2^{*}}\quad \text {for all } x\in {\mathbb {R}}^{N}. \end{aligned}$$

(5.2)

Using (5.1) and (5.2), we can obtain that

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}\vert \nabla \vert v_{n}\vert \vert ^{2}\eta ^{2}v_{L, n}^{2(\beta -1)}dx\le 2\int _{{\mathbb {R}}^{N}} \eta v_{L, n}^{2(\beta -1)}\vert v_{n}\vert \vert \nabla \vert v_{n}\vert \vert \vert \nabla \eta \vert dx +C\int _{{\mathbb {R}}^{N}} \eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2^{*}}dx.\nonumber \\ \end{aligned}$$

(5.3)

For each $\delta >0$, using Young’s inequality, we have from (5.3) that

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}\vert \nabla \vert v_{n}\vert \vert ^{2}\eta ^{2}v_{L, n}^{2(\beta -1)}dx\le & {} 2\delta \int _{{\mathbb {R}}^{N}} \eta ^{2}v_{L, n}^{2(\beta -1)} \vert \nabla \vert v_{n}\vert \vert ^{2}dx+2C_{\delta }\int _{{\mathbb {R}}^{N}} v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx\\&+C\int _{{\mathbb {R}}^N}\eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2^{*}}dx. \end{aligned}$$

Choosing $\delta \in (0, \frac{1}{4})$, it yields

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}\vert \nabla \vert v_{n}\vert \vert ^{2}\eta ^{2}v_{L, n}^{2(\beta -1)}dx\le C\int _{{\mathbb {R}}^{N}} v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx+C\int _{{\mathbb {R}}^{N}} \eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2^{*}}dx.\nonumber \\ \end{aligned}$$

(5.4)

On the other hand, by the Sobolev and Hölder inequalities, we have

$$\begin{aligned} \vert \omega _{L, n}\vert _{2^{*}}^{2}\le & {} C\int _{{\mathbb {R}}^{N}} \vert \nabla \omega _{L, n}\vert ^{2}dx=C\int _{{\mathbb {R}}^{N}} \vert \nabla \left( \eta \vert v_{n}\vert v_{L, n}^{\beta -1}\right) \vert ^{2}dx\nonumber \\\le & {} C\beta ^{2}\Big (\int _{{\mathbb {R}}^{N}} v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx+\int _{{\mathbb {R}}^{N}} \eta ^{2}v_{L, n}^{2(\beta -1)}\vert \nabla \vert v_{n}\vert \vert ^{2}dx\Big ). \end{aligned}$$

(5.5)

Combining (5.4) and (5.5), we have

$$\begin{aligned} \vert \omega _{L, n}\vert _{2^{*}}^{2} \le C\beta ^{2}\Big (\int _{{\mathbb {R}}^{N}} v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx+\int _{{\mathbb {R}}^{N}}\eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2^{*}}dx\Big ). \end{aligned}$$

(5.6)

Let $\beta =\frac{2^{*}}{2}$, by the definition of $\omega _{L, n}$ and (5.6), we rewrite the last inequality as

$$\begin{aligned}&\Big (\int _{{\mathbb {R}}^{N}} (\eta \vert v_{n}\vert v_{L, n}^{(2^{*}-2)/2})^{2^{*}}\Big )^{2/2^{*}} \\&\quad \le C(N, 2)\Big \{\Big (\int _{{\mathbb {R}}^{N}}(\eta \vert v_{n}\vert v_{L, n}^{(2^{*}-2)/2} )^{2^{*}}dx\Big )^{2/2^{*}}\Big (\int _{\vert x\vert \ge R-r} \vert v_{n}\vert ^{2^{*}}\Big )^{(2^{*}-2)/2}\\&\qquad +\int _{{\mathbb {R}}^{N}} v_{L, n}^{2^{*}-2}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx\Big \}\\&\quad \le C(N, 2)\Big \{\Big (\int _{{\mathbb {R}}^{N}}(\eta \vert v_{n}\vert v_{L, n}^{(2^{*}-2)/2} )^{2^{*}}dx\Big )^{2/2^{*}}\vert v_{n}\vert _{2^{*}(\vert x\vert \ge R/2)}^{2^{*}-2}\\&\qquad +\int _{{\mathbb {R}}^{N}} v_{L, n}^{2^{*}-2}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx\Big \}. \end{aligned}$$

From Lemma 4.5, $\vert v_{n}\vert \rightarrow \vert v\vert$ in $H^{1}({\mathbb {R}}^N)$, for R large enough, we conclude that

$$\begin{aligned} \vert v_{n}\vert _{2^{*}(\vert x\vert \ge R/2)}^{2^{*}-2}\le \frac{1}{2 C(N, 2)}\quad \text {uniformly in}\,\,n\in {\mathbb {N}}. \end{aligned}$$

Hence we obtain

$$\begin{aligned} \Big (\int _{\vert x\vert \ge R} (\vert v_{n}\vert v_{L, n}^{(2^{*}-2)/2})^{2^{*}}\Big )^{2/2^{*}}\le & {} 2C(N, 2)\int _{{\mathbb {R}}^{N}} v_{L, n}^{2^{*}-2}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx\\\le & {} \frac{C}{r^{2}}\int _{{\mathbb {R}}^{N}} \vert v_{n}\vert ^{2^{*}} dx. \end{aligned}$$

Using the Fatou’s lemma in the variable L, we have

$$\begin{aligned} \vert v_{n}\vert \in L^{{{2^{*}}^{2}}/2}(\vert x\vert \ge R)\quad \text {for}\,\, R\,\,\, \text {large enough}. \end{aligned}$$

(5.7)

Next, we note that if $\beta =2^{*}(t-1)/2t$ with $t={2^{*}}^{2}/2(2^{*}-2)$, then $\beta >1$ and $2t/(t-1)<2^{*}$. Now suppose that $\vert v_{n}\vert \in L^{2\beta t/(t-1)}(\vert x\vert \ge R-r)$ for some $\beta \ge 1$. Using the Hölder inequality with exponent $t/(t-1)$ and t, then (5.7) gives that

$$\begin{aligned} \vert \omega _{L, n}\vert _{2^{*}}^{2}\le & {} C\beta ^{2}\Big \{\Big (\int _{\vert x\vert \ge R-r}(\eta ^{2}\vert v_{n}\vert ^{2\beta })^{t/(t-1)}dx\Big )^{1-1/t}\Big (\int _{\vert x\vert \ge R-r} \vert v_{n}\vert ^{(2^{*}-2)t}\Big )^{1/t}\nonumber \\&+\frac{(R^{N}-(R-r)^{N})^{1/t}}{r^{2}}\Big (\int _{\vert x\vert \ge R-r}\vert v_{n}\vert ^{2\beta t/(t-1)} dx\Big )^{1-1/t}\Big \}\nonumber \\\le & {} C\beta ^{2}\Big (1+\frac{R^{N/t}}{r^{2}}\Big )\Big (\int _{\vert x\vert \ge R-r}\vert v_{n}\vert ^{2\beta t/(t-1)} dx\Big )^{1-1/t}. \end{aligned}$$

(5.8)

Letting $L\rightarrow +\infty$ in (5.8), we obtain

$$\begin{aligned} \vert v_{n}\vert _{2^{*}\beta (\vert x\vert \ge R)}^{2\beta }\le C\beta ^{2}\Big (1+\frac{R^{N/t}}{r^{2}}\Big )\vert v_{n}\vert _{2\beta t/(t-1)(\vert x\vert \ge R-r)}^{2\beta }. \end{aligned}$$

If we set $\chi :=2^{*}(t-1)/(2t)$, $s:=2t/(t-1)$, then

$$\begin{aligned} \vert v_{n}\vert _{\beta \chi s(\vert x\vert \ge R)}\le C^{1/\beta }\beta ^{1/\beta }\Big (1+\frac{R^{N/t}}{r^{2}}\Big )^{1/(2\beta )}\vert v_{n}\vert _{\beta s(\vert x\vert \ge R-r)}. \end{aligned}$$

(5.9)

Let $\beta =\chi ^{m}(m=1, 2, \ldots )$, we obtain

$$\begin{aligned} \vert v_{n}\vert _{\chi ^{m+1} s(\vert x\vert \ge R)}\le C^{\chi ^{-m}}\chi ^{m\chi ^{-m}}\Big (1+\frac{R^{N/t}}{r^{2}}\Big )^{1/(2\beta )}\vert v_{n}\vert _{\chi ^{m} s(\vert x\vert \ge R-r)}. \end{aligned}$$

It is clear that $2>N/t$. So if we take $r_{m}=2^{-(m+1)}R$, then (5.9) implies

$$\begin{aligned} \vert v_{n}\vert _{\chi ^{m+1} s(\vert x\vert \ge R)}\le & {} \vert v_{n}\vert _{\chi ^{m+1} s(\vert x\vert \ge R-r_{m+1})}\\\le & {} C^{\sum _{i=1}^{m}\chi ^{-i}}\chi ^{\sum _{i=1}^{m} i\chi ^{-i}}\exp \Big ( \sum _{i=1}^{m}\frac{\ln (1+2^{2(i+1)})}{2\chi ^{i}}\Big )\vert v_{n}\vert _{\chi s(\vert x\vert \ge R-r_{1})}\\\le & {} C\vert v_{n}\vert _{2^{*}(\vert x\vert \ge R/2)}. \end{aligned}$$

Letting $m\rightarrow \infty$ in the last inequality, we get

$$\begin{aligned} \vert v_{n}\vert _{L^{\infty }(\vert x\vert \ge R)}\le C\vert v_{n}\vert _{2^{*}(\vert x\vert \ge R/2)}. \end{aligned}$$

(5.10)

Using $\vert v_{n}\vert \rightarrow \vert v\vert$ in $H^{1}({\mathbb {R}}^N)$ again, for any fixed $a>0$, there exists $R>0$ such that $\vert v_{n}\vert _{L^{\infty }(\vert x\vert \ge R)}\le a$ for all $n\in {\mathbb {N}}$. Therefore, $\underset{\vert x\vert \rightarrow \infty }{\lim } \vert v_{n}(x)\vert =0$ uniformly in n.

To show that $\vert v_{n}\vert _{L^{\infty }({\mathbb {R}}^N)}<+\infty$, we need only show that for any $x_{0}\in {\mathbb {N}}$, there is a ball $B_{R}(x_{0})=\{x\in {\mathbb {R}}^N): \vert x-x_{0}\vert \le R\}$ such that $\vert v_{n}\vert _{L^{\infty }(B_{R}(x_{0}))}<+\infty$. We can use the same arguments and take $\eta \in C^{\infty }({\mathbb {R}}^N)$, $0\le \eta \le 1$ with $\eta (x)=1$ if $\vert x-x_{0}\vert \le \rho '$ and $\eta (x)=0$ if $\vert x-x_{0}\vert > 2\rho '$ and $\vert \nabla \eta \vert \le \frac{2}{\rho '}$, to prove that

$$\begin{aligned} \vert v_{n}\vert _{L^{\infty }(\vert x-x_{0}\vert \le \rho ')}\le C\vert v_{n}\vert _{2(\vert x\vert \ge 2\rho ')}. \end{aligned}$$

(5.11)

From (5.10) and (5.11), using a standard covering argument it follows that

$$\begin{aligned} \vert v_{n}\vert _{L^{\infty }({\mathbb {R}}^N)}\le C \end{aligned}$$

for some positive constant C.

For the case $N=2$, similar with the proof for the case $N\ge 3$, we also let $z_{L, n}:=\eta ^{2}v_{L, n}^{2(\beta -1)}v_{n}$ and $w_{L, n}:=\eta v_{L, n}^{\beta -1}\vert v_{n}\vert$ with $\beta >1$ to be determined later. Taking $z_{L, n}$ as a test function, we also have (5.1). Moreover, from the definition of g, for any $0<\zeta <V_{0}$ small, there exists $C_{\zeta }>0$ such that

$$\begin{aligned} g(x, t^{2})t^{2}\le \zeta t^{2}+C_{\zeta }\vert t\vert ^{q}\quad \text {for all } x\in {\mathbb {R}}^{N}. \end{aligned}$$

(5.12)

where $2<q<\infty$.

By (5.1) and (5.12), we obtain that

$$\begin{aligned} \int _{{\mathbb {R}}^{2}}\vert \nabla \vert v_{n}\vert \vert ^{2}\eta ^{2}v_{L, n}^{2(\beta -1)}dx\le 2\int _{{\mathbb {R}}^{2}} \eta v_{L, n}^{2(\beta -1)}\vert v_{n}\vert \vert \nabla \vert v_{n}\vert \vert \vert \nabla \eta \vert dx +C\int _{{\mathbb {R}}^{2}} \eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{q}dx.\nonumber \\ \end{aligned}$$

(5.13)

For any $\delta >0$, using Young’s inequality, we have from (5.13) that

$$\begin{aligned} \int _{{\mathbb {R}}^{2}}\vert \nabla \vert v_{n}\vert \vert ^{2}\eta ^{2}v_{L, n}^{2(\beta -1)}dx\le & {} 2\delta \int _{{\mathbb {R}}^{2}}\eta ^{2}v_{L, n}^{2(\beta -1)} \vert \nabla \vert v_{n}\vert \vert ^{2}dx+2C_{\delta }\int _{{\mathbb {R}}^{2}} v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx\\&+C\int _{{\mathbb {R}}^{2}}\eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{q}dx. \end{aligned}$$

Choosing $\delta \in (0, \frac{1}{4})$, it yields

$$\begin{aligned} \int _{{\mathbb {R}}^{2}}\vert \nabla \vert v_{n}\vert \vert ^{2}\eta ^{2}v_{L, n}^{2(\beta -1)}dx\le C\int _{{\mathbb {R}}^{2}} v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx+C\int _{{\mathbb {R}}^{2}} \eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{q}dx.\nonumber \\ \end{aligned}$$

(5.14)

On the other hand, by the Sobolev embedding,

$$\begin{aligned} \vert \omega _{L, n}\vert _{q}^{2}\le C\beta ^{2}\Big (\int _{{\mathbb {R}}^{2}} v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx+\int _{{\mathbb {R}}^{2}} \eta ^{2}v_{L, n}^{2(\beta -1)}\vert \nabla \vert v_{n}\vert \vert ^{2}dx \Big ). \end{aligned}$$

(5.15)

Using (5.14) and (5.15), we have

$$\begin{aligned} \vert \omega _{L, n}\vert _{q}^{2} \le C\beta ^{2}\Big (\int _{{\mathbb {R}}^{2}} v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx+\int _{{\mathbb {R}}^{2}} \eta ^{2}v_{L, n}^{2(\beta -1)}\vert v_{n}\vert ^{q}dx\Big ). \end{aligned}$$

(5.16)

Let $\beta =\frac{q}{2}$, by the definition of $\omega _{L, n}$ and (5.6), we rewrite the last inequality as

$$\begin{aligned} \Big (\int _{{\mathbb {R}}^{2}} (\eta \vert v_{n}\vert v_{L, n}^{(q-2)/2})^{q}\Big )^{2/q}\le & {} C(2, 2)\Big \{\Big (\int _{{\mathbb {R}}^{2}}(\eta \vert v_{n}\vert v_{L, n}^{(q-2)/2} )^{q}dx\Big )^{2/q}\Big (\int _{\vert x\vert \ge R-r} \vert v_{n}\vert ^{q}\Big )^{(q-2)/q}\\&+\int _{{\mathbb {R}}^{2}} v_{L, n}^{q-2}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx\Big \}\\\le & {} C(2, 2)\Big \{\Big (\int _{{\mathbb {R}}^{2}}(\eta \vert v_{n}\vert v_{L, n}^{(q-2)/2} )^{q}dx\Big )^{2/q}\vert v_{n}\vert _{q(\vert x\vert \ge R/2)}^{q-2}\\&+\int _{{\mathbb {R}}^{2}} v_{L, n}^{q-2}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx\Big \}. \end{aligned}$$

From Lemma 4.5, $\vert v_{n}\vert \rightarrow \vert v\vert$ in $H^{1}({\mathbb {R}}^2)$, we have $\vert v_{n}\vert \rightarrow \vert v\vert$ in $L^{q}({\mathbb {R}}^2)$. Thus, for R large enough, we conclude that

$$\begin{aligned} \vert v_{n}\vert _{q(\vert x\vert \ge R/2)}^{q-2}\le \frac{1}{2 C(2, 2)}\quad \text {uniformly in}\,\,n\in {\mathbb {N}}. \end{aligned}$$

Hence we obtain

$$\begin{aligned} \Big (\int _{\vert x\vert \ge R} (\vert v_{n}\vert v_{L, n}^{(q-2)/2})^{q}\Big )^{2/q}\le & {} 2C(2, 2)\Big (\int _{{\mathbb {R}}^{2}} v_{L, n}^{q-2}\vert v_{n}\vert ^{2} \vert \nabla \eta \vert ^{2} dx+\int _{{\mathbb {R}}^{2}} \eta ^{2} v_{L, n}^{q-2}\vert v_{n}\vert ^{2} dx\Big )\\\le & {} \frac{C}{r^{2}}\int _{{\mathbb {R}}^{2}} \vert v_{n}\vert ^{q} dx. \end{aligned}$$

Using the Fatou’s lemma in the variable L, we have

$$\begin{aligned} \vert v_{n}\vert \in L^{{{q}^{2}}/2}(\vert x\vert \ge R)\quad \text {for}\,\, R\,\,\, \text {large enough}. \end{aligned}$$

(5.17)

Next, we note that if $\beta =q(t-1)/2t$ with $t={q^{2}}/2(q-2)$, then $\beta >1$ and $2t/(t-1)<q$. Now suppose that $\vert v_{n}\vert \in L^{2\beta t/(t-1)}(\vert x\vert \ge R-r)$ for some $\beta \ge 1$. Using the Hölder inequality with exponent $t/(t-1)$ and t, then (5.16) gives that

$$\begin{aligned} \vert \omega _{L, n}\vert _{q}^{2}\le & {} C\beta ^{2}\Big \{\Big (\int _{\vert x\vert \ge R-r}(\vert v_{n}\vert ^{2\beta })^{t/(t-1)}dx\Big )^{1-1/t}\Big (\int _{\vert x\vert \ge R-r} \vert v_{n}\vert ^{(q-2)t}\Big )^{1/t}\nonumber \\&+\frac{(R^{2}-(R-r)^{2})^{1/t}}{r^{2}}\Big (\int _{\vert x\vert \ge R-r}\vert v_{n}\vert ^{2\beta t/(t-1)} dx\Big )^{1-1/t}\Big \}\nonumber \\\le & {} C\beta ^{2}\Big (1+\frac{R^{2/t}}{r^{2}}\Big )\Big (\int _{\vert x\vert \ge R-r}\vert v_{n}\vert ^{2\beta t/(t-1)} dx\Big )^{1-1/t}. \end{aligned}$$

(5.18)

Letting $L\rightarrow +\infty$ in (5.18), we obtain

$$\begin{aligned} \vert v_{n}\vert _{q\beta (\vert x\vert \ge R)}^{2\beta }\le C\beta ^{2}\Big (1+\frac{R^{2/t}}{r^{2}}\Big )\vert v_{n}\vert _{2\beta t/(t-1)(\vert x\vert \ge R-r)}^{2\beta }. \end{aligned}$$

If we set $\chi :=q(t-1)/(2t)$, $s:=2t/(t-1)$, then

$$\begin{aligned} \vert v_{n}\vert _{\beta \chi s(\vert x\vert \ge R)}\le C^{1/\beta }\beta ^{1/\beta }\Big (1+\frac{R^{2/t}}{r^{2}}\Big )^{1/(2\beta )}\vert v_{n}\vert _{\beta s(\vert x\vert \ge R-r)}. \end{aligned}$$

(5.19)

Let $\beta =\chi ^{m}(m=1, 2, \ldots )$, we obtain

$$\begin{aligned} \vert v_{n}\vert _{\chi ^{m+1} s(\vert x\vert \ge R)}\le C^{\chi ^{-m}}\chi ^{m\chi ^{-m}}\Big (1+\frac{R^{2/t}}{r^{2}}\Big )^{1/(2\beta )}\vert v_{n}\vert _{\chi ^{m} s(\vert x\vert \ge R-r)}. \end{aligned}$$

It is clear that $2>2/t$. So if we take $r_{m}=2^{-(m+1)}R$, then (5.19) implies

$$\begin{aligned} \vert v_{n}\vert _{\chi ^{m+1} s(\vert x\vert \ge R)}\le & {} \vert v_{n}\vert _{\chi ^{m+1} s(\vert x\vert \ge R-r_{m+1})}\\\le & {} C^{\sum _{i=1}^{m}\chi ^{-i}}\chi ^{\sum _{i=1}^{m} i\chi ^{-i}}\exp \Big ( \sum _{i=1}^{m}\frac{\ln (1+2^{2(i+1)})}{2\chi ^{i}}\Big )\vert v_{n}\vert _{\chi s(\vert x\vert \ge R-r_{1})}\\\le & {} C\vert v_{n}\vert _{q(\vert x\vert \ge R/2)}. \end{aligned}$$

Letting $m\rightarrow \infty$ in the last inequality, we get

$$\begin{aligned} \vert v_{n}\vert _{L^{\infty }(\vert x\vert \ge R)}\le C\vert v_{n}\vert _{q(\vert x\vert \ge R/2)}. \end{aligned}$$

(5.20)

Using $\vert v_{n}\vert \rightarrow \vert v\vert$ in $H^{1}({\mathbb {R}}^2)$ again, for any fixed $a>0$, there exists $R>0$ such that $\vert v_{n}\vert _{L^{\infty }(\vert x\vert \ge R)}\le a$ for all $n\in {\mathbb {N}}$. Therefore, $\underset{\vert x\vert \rightarrow \infty }{\lim } \vert v_{n}(x)\vert =0$ uniformly in n.

Similarly, in order to show that $\vert v_{n}\vert _{L^{\infty }({\mathbb {R}}^2)}<+\infty$, we need only show that for any $x_{0}\in {\mathbb {R}}^2$, there is a ball $B_{R}(x_{0})=\{x\in {\mathbb {R}}^2: \vert x-x_{0}\vert \le R\}$ such that $\vert \psi _{n}\vert _{L^{\infty }(B_{R}(x_{0}))}<+\infty$. We can use the same arguments and take $\eta \in C^{\infty }({\mathbb {R}}^2)$, $0\le \eta \le 1$ with $\eta (x)=1$ if $\vert x-x_{0}\vert \le \rho '$ and $\eta (x)=0$ if $\vert x-x_{0}\vert > 2\rho '$ and $\vert \nabla \eta \vert \le \frac{2}{\rho '}$, to prove that

$$\begin{aligned} \vert v_{n}\vert _{L^{\infty }(\vert x-x_{0}\vert \le \rho ')}\le C\vert v_{n}\vert _{2(\vert x\vert \ge 2\rho ')}. \end{aligned}$$

(5.21)

From (5.20) and (5.21), using a standard covering argument it follows that

$$\begin{aligned} \vert v_{n}\vert _{L^{\infty }({\mathbb {R}}^2)}\le C \end{aligned}$$

for some positive constant C and the proof is complete. $\square$

Now, we are ready to give the proof of Theorem 1.1.

Proof of Theorem 1.1

Let $\delta >0$ be such that $M_{\delta }\subset \Lambda$. We want to show that there exists ${\hat{\varepsilon }}_{\delta }>0$ such that for any $\varepsilon \in (0, {\hat{\varepsilon }}_{\delta })$ and any $u_\varepsilon \in \tilde{{\mathcal {N}}}_{\varepsilon }$ solution of problem (3.2), it holds

$$\begin{aligned} \Vert u_\varepsilon \Vert ^{2}_{L^{\infty }( \Lambda _{\varepsilon }^c)}\le a. \end{aligned}$$

(5.22)

We argue by contradiction and assume that there is a sequence $\varepsilon _{n}\rightarrow 0$ such that for every n there exists $u_{n}\in \tilde{{\mathcal {N}}}_{\varepsilon _{n}}$ which satisfies $J'_{\varepsilon _{n}}(u_{n})=0$ and

$$\begin{aligned} \Vert u_{n} \Vert ^{2}_{L^{\infty }( \Lambda _{\varepsilon _{n}}^c)}> a. \end{aligned}$$

(5.23)

Arguing as in Lemma 5.1, we have that $J_{\varepsilon _{n}}(u_{n})\rightarrow c_{V_{0}}$, and therefore we can use Proposition 4.1 to obtain a sequence $({\tilde{y}}_{n})\subset {\mathbb {R}}^{N}$ such that $y_n:=\varepsilon _{n}{\tilde{y}}_{n}\rightarrow y_{0}$ for some $y_{0}\in M$. Then, we can find $r>0$, such that $B_{r}(y_{n})\subset \Lambda$, and so $B_{r/\varepsilon _{n}}({\tilde{y}}_{n})\subset \Lambda _{\varepsilon _{n}}$ for all n large enough.

Using Lemma 5.1, there exists $R>0$ such that $|v_{n}|^{2}\le a$ in $B_R^c(0)$ and n large enough, where $v_{n}=u_{n}(\cdot +{\tilde{y}}_{n})$. Hence $\vert u_{n}\vert ^{2}\le a$ in $B_{R}^c({\tilde{y}}_{n})$ and n large enough. Moreover, if n is so large that $r/\varepsilon _{n} >R$, then $\Lambda _{\varepsilon _{n}}^c\subset B_{r/\varepsilon _{n}}^c({\tilde{y}}_{n})\subset B_{R}^c({\tilde{y}}_{n})$, which gives $\vert u_{n}\vert ^{2}\le a$ for any $x\in \Lambda ^c_{\varepsilon _{n}}$. This contradicts (5.23) and proves the claim.

Let now $\varepsilon _{\delta }:=\min \{{\hat{\varepsilon }}_{\delta }, {\tilde{\varepsilon }}_{\delta }\}$, where ${\tilde{\varepsilon }}_{\delta }>0$ is given by Theorem 4.1. Then we have $\text {cat}_{M_{\delta }}(M)$ nontrivial solutions to problem (3.2). If $u_\varepsilon \in \tilde{{\mathcal {N}}}_{\varepsilon }$ is one of these solutions, then, by (5.22) and the definition of g, we conclude that $u_\varepsilon$ is also a solution to problem (2.2).

Finally, we study the behavior of the maximum points of $\vert {\hat{u}}_{\varepsilon }\vert$, where ${\hat{u}}_{\varepsilon }(x):=u_{\varepsilon }(x/\varepsilon )$ is a solution to problem (1.4), as $\varepsilon \rightarrow 0^+$.

Take $\varepsilon _{n}\rightarrow 0^+$ and the sequence $(u_n)$ where each $u_n$ is a solution of (3.2) for $\varepsilon =\varepsilon _n$. From the definition of g, there exists $\gamma \in (0, a)$ such that

$$\begin{aligned} g(\varepsilon x, t^{2})t^{2}\le \frac{V_{0}}{K}t^{2},\quad \text {for all}\,\, x\in {\mathbb {R}}^{N},\, \vert t\vert \le \gamma . \end{aligned}$$

Arguing as above we can take $R>0$ such that, for n large enough,

$$\begin{aligned} \Vert u_n\Vert _{L^{\infty }(B_{R}^{c}({\tilde{y}}_{n}))}<\gamma . \end{aligned}$$

(5.24)

Up to a subsequence, we may also assume that for n large enough

$$\begin{aligned} \Vert u_n \Vert _{L^{\infty }(B_{R}({\tilde{y}}_{n}))}\ge \gamma . \end{aligned}$$

(5.25)

Indeed, if (5.25) does not hold, up to a subsequence, if necessary, we have $\Vert u_n \Vert _{\infty }<\gamma$. Thus, since $J'_{\varepsilon _{n}}(u_{\varepsilon _{n}})=0$, using (g5) and the diamagnetic inequality (2.1) that

$$\begin{aligned} \int _{{\mathbb {R}}^{N}}(\vert \nabla \vert u_n\vert \vert ^{2}+ V_{0}\vert u_n\vert ^{2})dx \le \int _{{\mathbb {R}}^{N}}g(\varepsilon _{n} x, \vert u_n\vert ^{2})\vert u_n\vert ^{2}dx \le \frac{V_{0}}{K}\int _{{\mathbb {R}}^{N}}\vert u_n\vert ^{2}dx \end{aligned}$$

and, being $K>2$, $\Vert u_n \Vert =0$, which is a contradiction.

Taking into account (5.24) and (5.25), we can infer that the global maximum points $p_{n}$ of $\vert u_{\varepsilon _{n}}\vert$ belongs to $B_{R}({\tilde{y}}_{n})$, that is $p_{n}=q_{n}+{\tilde{y}}_{n}$ for some $q_{n}\in B_{R}$. Recalling that the associated solution of problem (1.4) is ${\hat{u}}_{n}(x)=u_n(x/\varepsilon _{n})$, we can see that a maximum point $\eta _{\varepsilon _{n}}$ of $\vert {\hat{u}}_{n}\vert$ is $\eta _{\varepsilon _{n}}=\varepsilon _{n}{\tilde{y}}_{n}+\varepsilon _{n}q_{n}$. Since $q_{n}\in B_{R}$, $\varepsilon _{n}{\tilde{y}}_{n}\rightarrow y_{0}$ and $V(y_{0})=V_{0}$, the continuity of V allows to conclude that

$$\begin{aligned} \lim _{n}V(\eta _{\varepsilon _{n}})=V_{0}. \end{aligned}$$

The proof is now complete. $\square$