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

We study existence of semi-classical states for the nonlinear Choquard equation: -ε2Δv+V(x)v=1εα(Iα∗F(v))f(v)inRN,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\varepsilon ^2\Delta v+ V(x)v = {1\over \varepsilon ^\alpha }(I_\alpha *F(v))f(v) \quad \text {in}\ {\mathbb {R}}^N, \end{aligned}$$\end{document}where N≥3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N\ge 3$$\end{document}, α∈(0,N)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (0,N)$$\end{document}, Iα(x)=Aα/|x|N-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$I_\alpha (x)=A_\alpha /|{x}|^{N-\alpha }$$\end{document} is the Riesz potential, F∈C1(R,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F\in C^1({\mathbb {R}},{\mathbb {R}})$$\end{document}, F′(s)=f(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F'(s)=f(s)$$\end{document} and ε>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon >0$$\end{document} is a small parameter. We develop a new variational approach, in which our deformation flow is generated through a flow in an augmented space to get a suitable compactness property and to reflect the properties of the potential. Furthermore our flow keeps the size of the tails of the function small and it enables us to find a critical point without introducing a penalization term. We show the existence of a family of solutions concentrating to a local maximum or a saddle point of V(x)∈CN(RN,R)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V(x)\in C^N({\mathbb {R}}^N,{\mathbb {R}})$$\end{document} under general conditions on F(s). Our results extend the results by Moroz and Van Schaftingen (Calc Var Partial Differ Equ 52:199–235, 2015) for local minima (see also Cingolani and Tanaka (Rev Mat Iberoam 35(6):1885–1924, 2019)) and Wei and Winter (J Math Phys 50:012905, 2009) for non-degenerate critical points of the potential.


Introduction
In the recent years a large amount of papers has been devoted to investigate concentration phenomena of solutions to nonlinear Schrödinger equations with local sources around potential wells, namely local minima of some external potential functions. Starting to the celebrated papers by Floer and Weinstein [31] and Rabinowitz [58], several variational approaches were implemented and some efforts were done to obtain optimal results. We mention for instance [7, 16, 18-21, 29, 36, 37]. A more difficult problem seems to detect concentration phenomena around local maxima or saddle points of the potential type function. Some results are known for nonlinear Schrödinger equations under nondegeneracy conditions of the local maxima which allow to perform Lyapunov Schmidt reduction arguments [2,3,31,41,51]. More recently, del Pino and Felmer in [30] introduced a new reduction and proved a concentration result for solutions of nonlinear Schrödinger equation around local maxima and saddle points of the potential, assuming Ambrosetti-Rabinowitz type conditions and monotonicity conditions on the nonlinearity, which are crucial to apply a Nehari manifold approach. We refer to [28] for a generalization of the result of [30]. The more general result is contained in [8,9] where Byeon and the second author succeeded to show the existence of families of solutions to nonlinear Schrödinger equations with local nonlinearity of Berestycki-Lions type concentrating at critical points which are given by minimax method with suitable linking properties, e.g. local maxima, mountain pass critical points, non-degenerate critical points. See also [6,[10][11][12]39].
The goal of the present paper is to develop a new theoretical approach to obtain existence of solutions which concentrate at local maxima or saddle points of potential functions, under quite optimal assumptions on the nonlinearity and without any nondegeneracy conditions for class of nonlinear Schrödinger equations having local or nonlocal source.
As prototype of nonlocal problem in the source, we focus our analysis on the following class of equations (1.1) where ε > 0 is a small positive parameter, N ≥ 3, α ∈ (0, N ), is the Riesz potential, F(s) ∈ C 1 (R, R) and f (s) = F (s). We recall that in 1954 the Eq. (1.1) with N = 3, α = 2 and F(s) = 1 2 |s| 2 was introduced by Pekar [52] to describe the quantum theory of a polaron at rest. In 1976, (1.1) appeared in the work of Choquard on the modeling of an electron trapped in its own hole, in a certain approximation to the Hartree-Fock theory of plasma (see also [32]). More recently it has found a great attention due to models of self-gravitational collapse of a quantum mechanical wave function, proposed by Roger Penrose [53][54][55] and in that context it is known as as Schrödinger-Newton equation (see also [46,60]).
In literature, (1.1) is usually referred as nonlinear Choquard equation or Schrödinger equation with Hartree type potential. From a mathematical point of view, the early existence and symmetry results are due to Lieb [42] and Lions [43]. Successively, Ma and Zhao [44] classified all positive solutions to (1.1) for power nonlinearity and showed that they must be radially symmetric and monotonically decreasing about some fixed point. Recently Moroz and Van Schaftingen [48] investigated existence, some qualitative properties and decay asymptotics of positive ground state solutions to (1.1) for ε > 0 fixed when F satisfies the Berestycki-Lions type conditions. Other results are contained in [4,13,17,18,24,27,40,47,50,57].
In the present paper we are interested in the study the existence of concentrating family of solutions of (1.1) at local maxima or saddle point of V (x) as ε → 0.
Denoting u(x) = v(εx), the Eq. (1.1) is equivalent to Thus we try to find critical points of the corresponding functional: and we ask the existence of a concentrating family (u ε ) of solutions of (1.2) as ε → 0.
Firstly the concentration at nondegenerate critical points of the potential V (x) has been studied by Wei and Winter [62] using Lyapunov Schmidt reduction when N = 3, α = 2 and F(s) = s 2 . The case of local minima (possibly degenerate) of V when N = 3 and F(s) = s 2 has been considered in [22] by means of a penalization approach (see also [14,59,63]). More recently, Moroz and Van Schaftingen [49] proved existence of a single-peak solution of (1.1) concentrating at a local minima of V (x) for f (s) = |s| p−2 s, p ∈ [2, N +α N −2 ) via a new non-local penalization method. [64] extended the result in [49] and showed the existence under (f4) below, lim t→∞ They also proved the existence of multi-peak solutions, whose each peak concentrates at different local minimum of V (x) as ε → 0. We note that conditions p ≥ 2 or (1.3) is important in their arguments as it enables them to use linearized problems at infinity. See also [1,45,56] dealing with critical Choquard equations.
In [23] we developed a new variational approach which is applicable to a wide class of nonlinearities including F(s) = |s| p , p ∈ ( N +α N , N +α N −2 ). In particular, we can deal with the sublinear case p ∈ ( N +α N , 2), differently to [49]. We obtained the multiplicity of concentrating solutions via the cup-length of a critical set See also [38] for the effect of the topology of the potential wells on the existence of multi-bumps solutions.
The main purpose of this paper is to study the existence of concentrating family of solutions of nonlinear Choquard equation (1.1) at a local maximum or saddle point of V (x). To our knowledge, the only concentration result dealing nondegenerate local maxima is due to Wei and Winter [62], when N = 3, α = 2 and F(s) = s 2 .
The existence of concentrating families of solutions at local maxima and saddle points of V (x) is a more involved open problem and deformation argument using the standard gradient flow associated to I ε (u) does not seem enough. We also note that non-degeneracy of solutions of the limit problem − u is not known except the case N = 3, α = 2, F(u) = |u| 2 and it seems difficult to apply Lyapunov Schmidt reduction methods in general.
To show the existence of concentrating family of solutions, in this paper we develop a new deformation argument, which is partially inspired by [8,25,33,35].
Our deformation argument is developed for V (x) ∈ C 1 (R N , R) through a deformation in an augmented space R N × H 1 (R N ) and it has the following new features: (i) Our deformation flow is developed through a deformation for an augmented functional: We use the following translation of u ∈ H 1 (R N ) as a part of our new deformation argument: Thus, if ∇V ( p 0 ) = 0, choosing h = ∇V ( p 0 ), the traslation flow (1.4) gives a decreasing flow for I ε (u) in a small neighborhood of u ε . Thus ∇V ( p 0 ) gives a useful information for deformation argument. However we note that in We also note that the standard deformation flow η(t) : ) and the projection π ε . In the following sections, first we construct a deformation flow η for the augmented functional J ε (z, u) in R N × H 1 (R N ) and we construct a deformation flow for I ε (u) as a composition (π ε • η)(t). We also note that our new construction of a deformation flow works under weaker version of Palais-Smale type condition (see Proposition 4.5, 4.7 and 6.1). (ii) Another new aspect of our deformation flow is that it keeps the size of the tail of functions small during deformation. That is, defining the size of a tail of a function u by is the "center of mass" of u which will be defined in Sect. 3.3. We observe that for small κ ε with κ ε → 0, the set {u : T ε (u) ≤ κ ε } is positively invariant under our deformation flow. See Proposition 6.1 and (6.3) in Sect. 6. This property ensures that if u(x) concentrates around the center β(u) of mass, deformed function η(t, u) continues to concentrate around the center β(η(t, u)) of mass of the deformed functions η(t, u). The standard deformation flow does not have this property. Such a property is usually obtained by using tail minimization methods for local problems, that is, we solve the elliptic boundary problem outside of a large ball centered at β(u). We note that such a tail minimizing problem requires the unique solvability of the elliptic boundary problem and usually it is ensured for local problems, i.e., for nonlinear Schrödinger equations, under the condition f ∈ C 1 . For non-local problems, e.g. nonlinear Choquard equations such an approach does not work because of non-local feature of the problem. In Sects. 5 and 6 we develop a new deformation method in which the deformation flow is constructed through a deformation in an augmented space Our deformation method works for both of local and non-local problems. In a paper in preparation, we aim to apply this new approach to fractional problem (see [15] for concentration around local minima). See Remark 8.3 in Sect. 8 for an application to local problem (see also [26]).

Remark 1.1
In [8,9], a related deformation argument is developed for nonlinear Schrödinger equation: in a different way. Namely it is constructed as an iteration of 3 flows: is a pseudo-gradient vector associated to the functional corresponding to (1.2).
solution of the exterior problem: The procedure is rather complicated and in present paper we give an "easier" deformation argument through a construction flow in an augmented space R N × H 1 (R N ). We note that the exterior problem (1.6) is well-defined for local problem (1.5). But for nonlocal problem (1.2), the exterior problem is not well-defined because of non-locality of the problem.
To state our existence result for (1.2), we assume (f4) f (s) is odd and f is positive on (0, ∞).
We remark that the conditions (f1)-(f4) are in the spirit of Berestycki and Lions [5,34,48] and in our previous work [23] for a continuous potential V (x) we studied concentration at a local minimum under these conditions.
In the present paper we require much regularity on the potential V (x). Precisely for V (x) we assume We mainly study two situations where V (x) has a local maximum in or V (x) has a mountain pass geometry in . More precisely, we assume (LM) or (MP) below.
(MP) There exist e 0 , e 1 ∈ such that setting where n(x) ∈ R N is the unit outer normal at x ∈ ∂ .
We note that under the assumption (i), (ii) it is standard to see that V 0 is a critical value of V (x). Our main result is That is, there exist ε 0 > 0 and a family (u ε ) ε∈(0,ε 0 ] of solutions of (1.2) with the following property: for any sequence (ε j ) ∞ j=1 ⊂ (0, ε 0 ] with ε j → 0 after extracting a subsequence-we denote it by ε j for simplicity of notation-, there exist ( In (V1)-(V3), the assumption V (x) ∈ C N (R N , R) is used in order to show via Sard's Theorem that the set of critical values of V (x) is of measure 0. For a potential V (x) of class C 1 , we can show the existence of a solution under the following assumption of isolatedness of critical points of V (x) Namely we have This paper is organized as follows: In Sect. 2 we give some preliminary results. In Sect. 3 we study the limit problem. We introduce a Pohozaev type function P a (u) and a center β(u) of mass, which are used in this paper repeatedly. In Sect. 4 we introduce a neighborhood of expected solutions and we show a concentration-compactness type results for functional I ε (u). We will develop a local deformation argument in this neighborhood in Sects. 5, 6, and 7. Here newly introduced ε-dependent distance dist ε (·, ·) in H 1 (R N ) plays an important role. In Sect. 5 we introduce a functional T ε (u) to estimate the size of the tail of functions u and we construct a vector field, which decreases both of T ε (u) and I ε (u) and which enables us to generate a special deformation flow that keeps the tail of functions small. In Sect. 6 we give our new deformation result for I ε (u), which has new features stated above. Finally we give a proof of our main existence result in Sect. 7. In Sect. 8 we give a remark on concentration at a local minimum of V (x).

Preliminaries
In what follows, we use notation: for u ∈ H 1 (R N ) We also use notation for p ∈ R N ,

Estimates for Non-local Term
First we give some estimates for R N (I α * f )g and For proofs, we refer to [23]. We denote various constants, which are independent of u, by C, C , C , · · · Lemma 2.1 (c.f. Section 2.1 of [23]).
(i) For p, r > 1 and α ∈ (0, N ) with 1 In (ii), D L is given by where q satisfies 1 p + 1 q + 1 r = 2, in particular q > N N −α and I L α (x) is defined by We also have In particular, I ε (u) has mountain pass geometry uniformly in ε ∈ (0, 1] and we have Corollary 2.2 There exist ρ 0 > 0 and c 0 > 0 such that for ε ∈ (0, 1] We will use the following inequalities frequently: In fact, We can show the second inequality in a similar way.
Here D L > 0 is given in Lemma 2.1. In particular D L → 0 as L → ∞.
Proof We set We also set for i = 1, 2, 3 Since We can see that (i) holds. We can show (ii) in a similar way.
The above lemma gives a useful localization property of D(u). Finally in this section we give the following lemma on the behavior of bounded Palais-Smale sequences, which will help us to get concentration-compactness type result in Sect. 4.

Lemma 2.4 There exists
Extracting a subsequence if necessary, we may assume that for any L ∈ N there exists where a L is independent of j and satisfies a L → 0 as L → ∞. Here we apply Lemma 2.3 (i) with R = Lk L and L. Thus we have Let ρ 0 > 0 be the number given in Corollary 2.2. Since u (L) 0

Limit Problems
For a > 0 we define which appears as a limit equation for (1.2). That is, for a family (u ε (x)) of solutions of (1.2) and ( , that is, a solution of (3.1) with a = V (x 0 ). We denote by E a the least energy level for (3.1): In [48], the existence of a least energy solution is proved under the conditions (f1)-(f3) and They also proved that under (f1)-(f3), (f4') every ground state solution of (3.1) is radially symmetric with respect to some point in R N . It is also shown that any solution of (3.1) satisfies the Pohozaev identity: The least energy level E a is characterized as For c > 0 we set Arguing as in [48], we can show that

Scaling Argument for L a (u)
As in [23], to see the scaling property of the limit function L a (u), we consider for We have In particular, we have

Center of Mass
Here we introduce a center of mass β(u) in a neighborhood of a shifted compact set. We will use the following Then there exist ρ 2 > 0, R 0 > 0 and C 1 -function β : A similar center of mass is given in [8,9], which is locally Lipschitz continuous. Here we modify and improve the argument in [8,9] and give a center of mass β(u), which is of class C 1 .
We set for q ∈ R N and u ∈ D r * /6 We set Then we have

Finally we prove (iv). We set
In the following sections, we develop a deformation argument for I ε (u) in D ρ 2 for a suitable choice of D.

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

A Neighborhood Ä of Concentrating Points
In this section, we show that we may assume the following (V4) in addition to (V1)-(V3) and (LM) (or (MP)).
In fact, since E a is a continuous function of a ∈ (0, ∞), there exists α > 0 such that On the other hand, since V (x) is of class C N , the set of critical values of V (x) is of measure 0 in R by Sard Theorem. Therefore we may assume V 0 − α is a regular value of V (x). We set We observe that if V (x) satisfies (LM) ((MP) respectively) in , then V (x) satisfies (LM) ((MP) respectively) in α . We show just for (MP).
We may assume V (e 0 ), V (e 1 ) < V 0 − α. We set Then we can easily see that Since M 0 , M 1 are compact, we may assume after extracting a subsequence Choose e 0 , e 1 ∈ α so that e 0 is close to e 0 and e 1 is close to e 1 . Replacing , e 0 , e 1 , with α , e 0 , e 1 and = {c( we can see that (MP) holds.

A Neighborhood of Expected Solutions
In what follows, we assume (V1)-(V4) hold for and V 0 is a critical value of V (x) in . We write b = E V 0 and set We note that We remark that L(z, u) appears as a limit functional for I ε (u). In fact, for z ∈ R N and u(x) ∈ H 1 (R N ), we have In what follows, we denote the projections to the first and second components by and ω is not a least energy solution of L V (ξ ) (·).
We set Q = [0, 1] N and For ε > 0 we set and we try to find a critical point of I ε (u) in a neighborhood of K (ε) b . We introduce K b and K b to obtain necessary compactness properties, in particular, to show Proposition 4.5 below.
For our minimax argument, we also introduce It holds By (V1)-(V4) and Lemma 3.1, we see Here and in what follows we indicate compact sets by ·.
We set for ρ > 0 These sets are uniformly bounded with respect to ε ∈ (0, 1] and we have In what follows, for suitable 0 < ρ < ρ we develop a deformation argument in A (ε) ρ to find a critical point in N (ε) ρ .

Remark 4.4 The reason we introduce A
(ε) ρ is to construct neighborhoods which are suitable for our deformation arguments. Our neighborhood A (ε) ρ includes a suitable initial path in H 1 (R N ) which is related to a minimax argument in ⊂ R N . See Sect. 7.1 below. Our another neighborhood N (ε) ρ is precisely an ε-neighborhood of expected solutions with the profile in K b .

Concentration-Compactness Type Results
In this section we give an ε-dependent concentration-compactness type results, which will be useful to develop deformation theory in Sect. 6.

Proposition 4.5 There exists
In particular, for any ρ > 0 there exists j ρ ∈ N such that

Remark 4.6
To show the existence of a family concentrating at a local minimum of V (x), in [23] we obtained a similar result for (u j ) ∞ j=1 ⊂ N (ε j ) ρ 3 but without the assumption (4.10). To study concentration at local maxima and saddle points, we need (4.10).
Step 4: For ρ > 0 small, dist ε j (u j , K (ε j ) b ) → 0 It is clear that ξ j = ξ j + h j is in a ρ -neighborhood of and thus so is ξ 0 . Since Thus choosing ρ 3 > 0 small, the proof is completed.

Proposition 4.7 Let ρ 1 > 0 be the number given in Lemma 2.4. For ε ∈ (0, 1] fixed, I ε (u) satisfies the Palais-Smale type condition in A
then (u j ) ∞ j=1 has a strongly convergent subsequence in H 1 (R N ). Moreover, after extracting a subsequence if necessary, assume u j → u 0 strongly as j → ∞. Then u 0 satisfies I ε (u 0 ) = 0 and Extracting a subsequence if necessary, we may assume for some

Using Lemma 2.4 and arguing as in
Step 1 of the proof of Proposition 4.5, we have the strong convergence of (u j ). (4.19) follows from H ε (u j ) → 0.

A Choice of Neighborhoods and Gradient Estimates
We choose ρ * * > 0 small so that in a neighborhood A (ε) ρ * * of K (ε) b , we can develop a deformation argument for a proof of our main result.
We set Here S b is defined in (4.1). Applying the argument in Sect. 3.3 with D = S b , D = S b and D ρ = S b,ρ , we can define the center of mass: We choose and fix ρ * , ρ * * > 0 such that where ρ 2 is given above and ρ 0 (ρ 1 , ρ 3 respectively) is given in Corollary 2.2 (Lemma 2.4, Proposition 4.5 respectively). We will use relation 16ρ * < ρ * * later in the proof of Lemma 6.9. We note that the center of mass β(u) is defined on A (ε) In fact, by the definition of A Thus by Proposition 3.3 (i) we have (4.21). By Propositions 4.5 and 4.7, we have the following estimates.
Then there exists ν ε > 0 such that In what follows we assume without loss of generality ν ε ≤ ν 0 .
In the following Sect. 5, we develop a special deformation argument for I ε (u).

Functional T (u)
To find a critical point of I ε (u) in a neighborhood N (ε) ρ of expected solutions, it is important to control the size of u outside of a ball B(β(u), 4 √ ε ).
We set for u ∈ S b,ρ 2 and ε > 0 We note that T ε (u) is translation invariant, that is, We use T ε (u) to estimate the size of u outside of a ball B(β(u), 4 √ ε ). In this section, we extend our idea in [23] to generate a special deformation flow for I ε (u), which keeps T ε (u) small along the flow.

A Special Vector Field in A ( ) * *
To construct a deformation flow which keeps the size of tail T ε (u) small, we find a special vector field in this section.
We note A (ε) ρ * * is bounded and so there exists C > 0 such that First we decompose u ∈ A (ε) ρ * * into a center part u (1) and a tail part u (2) . We denote the integer part of a > 0 by [a]. Since there exists k ∈ {1, 2, · · · , [ε −1/4 ] − 1} such that In what follows we denote by c ε various constants which do not depend on u and satisfy c ε → 0 as ε → 0. We set where ζ R (x), ζ R (x) are defined in (2.1). We also set These function also give decomposition of u into a center part and a tail part. Clearly we We note that u (1) , u (2) , M 1 (u), M 2 (u) depend on ε. But for simplicity of notation, we omit ε from the notation.
We use −u (2) to construct a deformation flow and we use M 1 (u) and M 2 (u) to estimate effects of −u (2) .
u (2) has the following properties.

11)
T ε (u)u (2) = 2T ε (u). (5.12) (v) For c 0 > 0 given in Corollary 2.2, we have From Lemma 5.1, we can observe a vector field u → −u (2) has good properties for deformation. By (ii), (iii), −u (2) does not effect the center part M 1 (u) and the center β(u) of mass of u. By (5.12) and (5.13), −u (2) gives a direction which decreases both of I ε (u) and T ε (u) provided T ε (u) ≥ c ε c 0 . Thus it is convenient to construct a deformation flow for I ε (u) which keeps the size T ε (u) of tail small.
Choice of κ ε . By the compactness of S b , we have For c ε > 0 given in Lemma 5.1, we set With this choice of κ ε , we have the following corollary. In what follows, we use the following notation for c ∈ R
For later use, we state the following lemma, which states that the property u ∈ [T ε ≤ κ ε ] ensures that u concentrates around the center of mass β(u).
Here c ε > 0 is independent of u and satisfies c ε → 0 as ε → 0.
We note that H ε (u) gives a useful information on deformation. In fact, for h ∈ R N we have Thus, if H ε (u) = 0, the translation flow: gives a decreasing flow in a neighborhood of u.
The property (v) means that the set [T ε ≤ κ ε ] is positively invariant for the flow η(t, u), i.e., This property is related to the tail minimizing flow developed in [23]. In [23], we used the tail minimizing flow separately from the deformation flow (the steepest descent flow) for I ε (u). Here, extending the idea in [23] we construct a deformation flow for I ε (u) which keeps the size T ε (u) of the tail u| R N \B(β(u),4/ √ ε) small.
In the following sections, replacing scaling (6.4) to translation (6.2), we give a proof of Proposition 6.1.

Augmented Functional
To prove Proposition 6.1, we consider the following functional in the augmented space Recalling D = (∂ z , ∂ u ), we have As in Corollary 2.2, we have Corollary 6.4 There exist ρ 0 > 0 and c 0 > 0 such that To show our Proposition 6.1, we develop a deformation argument in R N × H 1 (R N ) and we construct a flow η(t, u) through a flow η(t, z, u) on a product space We introduce a pseudo-distance DIST ε (·, ·) on R N × H 1 (R N ), which is related to dist ε (·, ·), by DIST ε ((z, u), (z , u ) We set Clearly these sets are uniformly bounded with respect to ε ∈ (0, 1] and we have N ρ . From Proposition 4.9 (i), Corollary 5.3 and Lemma 6.3 we have the following Proposition 6.5 Let 0 < ρ * < ρ * * be the numbers satisfying (4.20). Then we have (i) There exist ν 0 > 0 and δ 0 > 0 independent of ε such that for ε > 0 small (ii) Suppose that (6.1) holds, in other words, it holds that Then there exists ν ε > 0 such that We note that we may assume ν ε < ν 0 .

Construction of a Vector Field
In what follows, we will show that the existence of a critical point . Arguing indirectly, we assume (6.1) holds. To construct a deformation flow, we find a special vector field V z,u : . Since (6.5) and (6.7) hold by Proposition 6.5, for (z, We compute for (z, u) ∈ A (ε) ρ * * and ≥ 0 (6.11) where C 1 > 0 is independent of ε and u. Here we used (5.12) and the boundedness of For κ ε defined in (5.15), we set Then we have Proposition 6.6 Suppose that (6.1) holds. Then for ε ∈ (0, 1 (6.14) In the above proposition, we write In particular, We use similar formulas also for M 1 (u) and M 2 (u) 2 H 1 .
Proposition 6.7 Suppose that (6.1) holds. Then for ε > 0 small, there exists a locally Lipschitz vector field W (z, u) : u be a vector field given in Proposition 6.6. We remark that for any We may choose a neighborhood U z,u of (z, u) so that Using a partition of unity, we can construct a locally Lipschitz continuous vector field in a standard way. We can easily see that W (z, u) satisfies (i)-(iv).
We note that W (z, u) is bounded in the following sense: for all (z, u), where C > 0 is independent of ε, (z, u).
We compute Thus, we have from Proposition 6.7 that (6.20)

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

Existence Under the Condition (MP)
First we consider ( Proof of Proposition 7.1 Let e 1 , e 2 , be given in (MP). We may choose ρ * > 0 smaller if necessary and choose ρ * * > 0 so that (4.20) holds.
Next we will show under (7.5)-(7.6) and (7.11) that γ ε (s, ξ) satisfies We note that (7.13) is incompatible with (7.12) and it shows the existence of a critical (2) Set then e 0 and e 1 belong to different components of W .
Step 4: Conclusion. (7.12) and (7.13) are incompatible and thus (6.1) does not hold. Thus we have the conclusion of Proposition 7.1.

Existence Under the Condition (LM)
In this section we consider ( Proof of Proposition 7. 2 Let ω 0 (x) be a least energy solution corresponding to b = E V 0 . We choose s 0 ∈ (0, 1 2 ) satisfying (7.1) and set γ 0ε (s, ξ) : We note that Thus there exists δ > 0 such that Moreover for any δ ∈ (0, δ) we have for sufficiently small ε > 0 We also note that γ 0ε (s, ξ) Arguing as in the proof of Proposition 7.1, we can prove Proposition 7.2.

End of the Proof of Theorem 1.2
Finally we derive our Theorem 1.2 from Propositions 7.1 and 7.2.
End of the proof of Theorem 1.2 Let V 0 be the critical value given by (MP) or (LM).

Potential V(x) of Class C 1
In previous sections we consider the situation where the set of critical values {V (x) : x ∈ , ∇V (x) = 0} is of measure 0, which is ensured by Sard Theorem for V (x) ∈ C N (R N , R). In this section we assume just V (x) ∈ C 1 (R N , R). Then the set of critical values may not be of measure 0.
We have the following weaker result.