Multiple semiclassical solutions for a nonlinear Choquard equation with magnetic field

Abstract

In the present paper, we study the existence of multiple solutions for a nonlinear Choquard equation in the presence of a magnetic field. Using variational methods, penalization techniques and Ljusternik–Schnirelmann theory, we relate the number of solutions with the topology of the set where the potential attains its minimum value.

Keywords

Choquard equation magnetic field variational methods penalization techniques Ljusternik–Schnirelmann theory

1. Introduction

In this paper we study the existence of multiple solutions the nonlinear complex Choquard equation $\begin{matrix} (P_{ε}) & \{\begin{matrix} {(\frac{ε}{i} \nabla - A (z))}^{2} u + V (z) u = ε^{μ - N} (\frac{1}{| z |^{μ}} * F (| u |^{2})) f (| u |^{2}) u, z \in R^{N}, \\ u \in H^{1} (R^{N}, C), \end{matrix} \end{matrix}$ where $N ⩾ 2$ , $0 < μ < 2$ , $F (s) = \int_{0}^{s} f (t) d t$ and i is the imaginary unit. The function $A : R^{N} \to R^{N}$ denotes a continuous magnetic potential associated with a magnetic field B (i.e. $curl A = B$ ) and the operator $\begin{matrix} {(\frac{ε}{i} \nabla - A (z))}^{2} \end{matrix}$ is defined by $\begin{matrix} {(\frac{ε}{i} \nabla - A (z))}^{2} ψ : = - ε^{2} Δ ψ - \frac{2 ε}{i} A \cdot \nabla ψ + | A |^{2} ψ - \frac{ε}{i} ψ div A . \end{matrix}$ The function V is a real electric potential and the nonlinear term f is a superlinear function. In many physical applications, nonlocal Hartree type nonlinearities appear naturally. In this case the bosons may condense into a spatially localized cluster that, in the mean-field regime, is described by a one-particle wave function corresponding to a solitary wave solution or soliton of the nonlinear Hartree equation in $R^{N}$ . Here, the function $\frac{1}{| z |^{μ}}$ is usually called the Riesz potential and it appears naturally in the propagation of electromagnetic waves in plasmas [8] and accounts for the finite-range many-body interaction in the theory of Bose–Einstein condensation [18].

There have been a great deal of interests in studying the existence, multiplicity and qualitative property of standing wave solutions for the nonlinear Schrödinger equation without magnetic field, i.e. $A = 0$ . If the interaction between the particles is omitted, then Eq. ( P ε ) becomes a standard local Schrödinger equation like $\begin{matrix} (1.1) & - ε^{2} Δ u + V (x) u = f (u), x \in R^{N} . \end{matrix}$ For the above equation, the geometry of the potential V affects the existence of solutions. Next, we will make a brief review about this subject. Assuming that V satisfying ${inf}_{x \in R^{N}} V (x) > 0$ is a globally bounded potential with a nondegenerate critical point, Floer and Weinstein [23] studied firstly the existence of single and multiple spike solutions for (1.1) based on a Lyapunov–Schmidt reduction for the case where $N = 1$ and f is a cubic nonlinearity. Since then, many mathematicians have been interested in the existence and the concentration of solutions of Eq. (1.1) under various assumptions on the potential V. In [5], Ambrosetti et al. studied the problem with polynomial degenerate potential V. In [38], Rabinowitz proved the existence of a positive ground state for any $ε > 0$ assuming that $\begin{matrix} (1.2) & 0 < a ⩽ V (x) ⩽ \underset{| x | \to + \infty}{lim inf} V (x) for all x \in R^{N} and some a > 0, \end{matrix}$ with strict inequality on a set of positive measure. Using a local variational approach, del Pino and Felmer [19] constructed positive solutions by supposing that there is a bounded open set $Λ \subset R^{3}$ such that $\begin{matrix} V_{0} : = inf_{x \in Λ} V (x) < inf_{x \in \partial Λ} V (x), \end{matrix}$ then if ${inf}_{R^{N}} V (x) > 0$ there also exists a positive semiclassical solution. We refer the readers to [6,10,20,21,35] and references therein for recent research progress involving the nonlinear Schrödinger equation.

If the interaction between the particles is considered, then we are lead to the nonlocal Schrödinger equation $\begin{matrix} (1.3) & - ε^{2} Δ u + V (x) u = (\int_{R^{3}} \frac{u^{p} (y)}{| x - y |^{μ}} d y) u^{p - 1}, x \in R^{N} . \end{matrix}$ For the case $ε = 1$ , $p = 2$ and $V > 0$ , this problem is usually called Choquard equation and has been investigated in the nonperiodic case by many authors, cf. [28–30] and the references therein. If the potential $V (x)$ is periodic in $x_{i}$ , $i = 1, 2, \dots, N$ and sign-changing, assuming 0 is in the gap of the spectrum of $- Δ + V (x)$ , Buffoni et al. in [9] firstly obtained the existence of one nontrivial solution. Ackermann [1] proposed an approach to prove the existence of infinitely many geometrically distinct weak solutions for a general class of response function and nonlinearities. Involving the properties of the ground state solutions, Ma and Zhao [32] have considered Eq. (1.3) for $p ⩾ 2$ , proving that every positive solution is radially symmetric and monotone decreasing about some point, under the assumption that a certain set of real numbers, defined in terms of N, α and q, is nonempty. Under the same assumption, Cingolani, Clapp and Secchi [13] have studied the existence and multiplicity of solutions in the electromagnetic case, showing the regularity and behavior at infinity of the ground states of (1.3). Moroz and Van Schaftingen [33] have eliminated this restriction, showing the regularity, positivity and radial symmetry of the ground states for the optimal range of parameters. The same authors, in [34], have studied the existence of ground states under the assumption of Berestycki–Lions type, while that in [36], they have obtained the existence of groundstates of nonlinear Choquard equations for a critical case in the sense of Hardy–Littlewood–Sobolev inequality.

However, there are not so many works about the existence of semiclassical solutions for Eq. (1.3), since little is known about the ground states of the corresponding autonomous limit problem $\begin{matrix} (1.4) & - Δ u + u = (\int_{R^{3}} \frac{u^{p} (y)}{| x - y |^{μ}} d y) u^{p - 1} \forall x \in R^{N} . \end{matrix}$ Very recently, for $p = 2$ , Lenzmann in (1.4), has obtained the nondegeneracy property of the ground state solution for [25]. The same property was also obtained by Wei and Winter [40] in order to study the existence of multi-bump solutions for a Schrödinger–Newton system with assumptions that ${inf}_{x \in R^{3}} V (x) > 0$ and $V \in C^{2} (R^{3})$ . In [42], Yang and Ding have considered the semiclassical problem for nonlocal problem with critical frequency and obtained the existence of solutions which go to 0 with suitable parameter p, μ. In [35], Moroz and Van Schaftingen proved the existence and concentration behavior of the semiclassical states for the problem with $F (u) = u^{p}$ for small ε. There, they have developed a penalization technique by constructing supersolutions to a linearization of the penalized problem in an outer domain and then estimate the solutions of the penalized problem by some comparison principle.

The appearance of the magnetic field also brings additional difficulties to the problem. For example, the effects of the magnetic fields on the linear spectral sets and on the solution structure, and the possible interactions between the magnetic fields and the linear potentials. An important point that we could point out is related to $L^{\infty}$ estimates, because we cannot use directly bootstrap arguments, see for example, the proof of Lemma 4.1. Therefore, equations having a magnetic field have been studied much less than equations which do not have a magnetic field. It seems that [22] was the first work which studied the existence of solutions of $\begin{matrix} (1.5) & {(- i \nabla + A (x))}^{2} u + V (x) u = | u |^{p - 2} u . \end{matrix}$ After, the existence and multiplicity of solutions of (1.5) were obtained in [7] under the assumptions that V, g and $B = curl A$ depend periodically on $x \in R^{N}$ . In [17], the authors have showed that the equation has infinitely many $2 π$ -periodic solutions. For the semiclassical case, existence and concentration phenomena of semiclassical solutions of ( P ε ) were studied in [24] with subcritical nonlinearities. There, the potential V satisfies ${inf}_{x \in R^{N}} V (x) > 0$ and other assumptions on V and B are made to assure that a priori compactness condition holds for any energy level and any $ε > 0$ . In [12], the multiplicity of solutions have obtained under the same assumptions on V and B. In [15], supposing that $A \in L^{\infty} \cap C^{1}$ , the existence of semiclassical states was proved for $\begin{matrix} (1.6) & {(- i ε \nabla + A (x))}^{2} u + V (x) u = K (x) | u |^{p - 2} u . \end{matrix}$ The authors also proved that these solutions really concentrates at the nondegenerate critical point of the function $V {(x)}^{θ} K {(x)}^{\frac{- 2}{p - 2}}$ , $θ = \frac{p}{p - 2} - \frac{N}{2}$ . For the semiclassical case, the existence and multiplicity of solutions were obtained in [21] under the key assumption that the set ${x : V (x) ⩽ 0}$ is nonempty and is of finite measure. In [4], the authors have considered the existence of multiplicity of semiclassical solutions via Ljusternik–Schnirelmann category theory. In [20], the authors have explored the existence and concentration phenomena of semiclassical solutions for the equation with magnetic field and critical nonlinearities. In [10], the authors have established the existence and uniqueness of multi-bump bound states. In [39], the author have studied the asymptotic behavior of the solutions to the nonlinear Schrödinger evolution equation with an external magnetic field A by taking as initial datum the ground state of an associated autonomous stationary equation.

As far as we know there are not so many papers deal with Hartree type of equation under the effect of a magnetic field. In [16], by using a penalization technique, the authors have been able to study the existence of multi-bump type solutions in the case of nonlocal nonlinearities of Hartree type.

It is therefore quite natural to ask how the presence of the magnetic field A and the external potential V will influence the existence of standing wave solutions of the Schrödinger equation ( P ε ) with nonlocal nonlinearity.

Inspired by recent works [4,16,42], in the present paper we study the existence and multiplicity of semiclassical states of the nonlinear Schrödinger equation under the effect of the magnetic potential and nonlocal nonlinearities.

Before stating our main result, we need to present the hypotheses on the potential V and the nonlinearity f. Here, we assume that

$V_{0} : = {inf}_{z \in R^{N}} V (z) > 0$ ;

there exists an open bounded set $Ω \subset R^{N}$ such that $\begin{matrix} V_{0} < min_{z \in \partial Ω} V (z) \end{matrix}$ and $M : = {z \in Ω : V (z) = V_{0}} \neq \emptyset$ .

We also suppose that

f \in C^{1} ([0, + \infty), R)

and it satisfies

${lim}_{s \to 0^{+}} f (s) = 0$ ;

there exists $q \in (2, 2^{*} (\frac{2 (N - μ)}{2 N}))$ such that $\begin{matrix} lim_{s \to \infty} \frac{f (s)}{s^{(q - 2) / 2}} = 0, \end{matrix}$ where $2^{*} : = 2 N / (N - 2)$ if $N ⩾ 3$ ;

$f^{'} (s) > 0$ for each $s > 0$ .

We will establish a relation between the number of solutions of ( P ε ) and the topology of the set M. In order to make a precise statement let us recall that, for any closed subset Y of a topological space X, the Ljusternik–Schnirelmann category of Y in X, ${cat}_{X} (Y)$ , stands for the least number of closed and contractible sets in X which cover Y.

The main result of this paper is the following:

Theorem 1.1.
Suppose that V satisfies $(V_{1})$ – $(V_{2})$ and f satisfies $(f_{1})$ – $(f_{3})$ . If $μ \in (0, 2)$ , then for any $δ > 0$ such that $\begin{matrix} M_{δ} : = {z \in R^{N} : dist (z, M) < δ} \subset Ω, \end{matrix}$ there exists $ε_{δ} > 0$ such that, for any $ε \in (0, ε_{δ})$ , the problem ( P ε ) has at least ${cat}_{M_{δ}} (M)$ weak solutions. Moreover, if $ε_{n} \to 0^{+}$ , $u_{ε_{n}}$ is one of these solutions and $η_{ε_{n}} \in R^{N}$ is a global maximum point of $| u_{ε_{n}} |$ , we have that $\begin{matrix} lim_{ε_{n} \to 0^{+}} V (η_{ε_{n}}) = V_{0} . \end{matrix}$

Note that the concentration result depends on maximum point of each solution and the minimum point of potential V. On the other hand, the existence result is true for all magnetic field A which is continuous. Hence, the concentration result does not depend on the continuous magnetic field A. Moreover, for simplicity, we treat only the case $N ⩾ 3$ because the case $N = 1, 2$ can be treated in a similar way, since we are working with polynomial type growth. The restriction on μ is related to the approach used, more precisely, the penalization technique explored in Section 2.

The paper is organized as follows. In Section 2, we show the variational framework and we define the auxiliary problem auxiliary problem. In Section 3, we prove a multiplicity result for the auxiliary problem, while that in Section 4 we prove the main result.
2. Variational framework

In this section, we fix some notations and present the variational setting of our problem. Throughout the paper we write only $\int u$ instead of $\int_{R^{N}} u (z) d z$ . For any set $B \subset R^{N}$ , we denote by $B^{c} : = R^{N} ∖ B$ the complement of B.

Using the change variable $v (x) = u (ε x)$ , it is easy to see that ( P ε ) is equivalent to $\begin{matrix} (D_{ε}) & \{\begin{matrix} {(\frac{1}{i} \nabla - A_{ε} (x))}^{2} v + V_{ε} (x) v = (\frac{1}{| x |^{μ}} * F (| v |^{2})) f (| v |^{2}) v, x \in R^{N}, \\ v \in H^{1} (R^{N}, C), \end{matrix} \end{matrix}$ with $A_{ε} (x) : = A (ε x)$ and $V_{ε} (x) : = V (ε x)$ .

Hereafter, we say that a function $u \in H_{ε}$ is a weak solution of the problem ( D ε ) if $\begin{matrix} Re (\int \nabla_{ε} u \overline{\nabla_{ε} v} + V_{ε} (x) u \overline{v} - (\frac{1}{| x |^{μ}} * F (| u |^{2})) f (| u |^{2}) u \overline{v}) = 0 for each v \in H_{ε}, \end{matrix}$ where $H_{ε} = H_{ε} (R^{N}, C)$ is the Hilbert space obtained by the closure of $C_{0}^{\infty} (R^{N}, C)$ under the scalar product $\begin{matrix} {⟨ u, v ⟩}_{ε} : = Re (\int \nabla_{ε} u \overline{\nabla_{ε} v} + V_{ε} (x) u \overline{v}), \end{matrix}$ the notation $Re (w)$ denotes the real part of $w \in C$ , $\overline{w}$ is its conjugate, $\nabla_{ε} u : = (D_{1}^{ε} u, D_{2}^{ε} u, \dots, D_{N}^{ε} u)$ and $D_{j}^{ε} : = i^{- 1} \partial_{j} - A_{j} (ε x)$ for $j = 1, \dots, N$ . The norm induced by this inner product is given by $\begin{matrix} ∥ u ∥_{ε} = {(\int | \nabla_{ε} u |^{2} + V_{ε} (x) | u |^{2})}^{1 / 2} . \end{matrix}$

As proved by Esteban and Lions in [22, Section II], for any $u \in H_{ε}$ there holds $\begin{matrix} (2.1) & | \nabla | u | (x) | = | Re (\nabla u \frac{\overline{u}}{| u |}) | = | Re ((\nabla u - i A_{ε} u) \frac{\overline{u}}{| u |}) | ⩽ | \nabla_{ε} u (x) | . \end{matrix}$ The above expression is called of diamagnetic inequality. The above inequality ensures that if $u \in H_{ε}$ , then $| u | \in H^{1} (R^{N}, R)$ . Furthermore, the embedding $H_{ε} ↪ L^{q} (R^{N}, R)$ is continuous for each $2 ⩽ q ⩽ 2^{*}$ and, for each bounded set $Λ \subset R^{N}$ and $2 ⩽ q < 2^{*}$ , the embedding below is compact $\begin{matrix} (2.2) & H_{ε} ↪ L^{q} (Λ, R) . \end{matrix}$

In order to use variational method, we must have $\begin{matrix} (2.3) & | \int (\frac{1}{| x |^{μ}} * F (| u |^{2})) F (| u |^{2}) | < \infty \forall u \in H_{ε} . \end{matrix}$ To show this, it is important to recall the Hardy–Littlewood–Sobolev inequality, which will be frequently used in the present paper.

Proposition 2.1 (Hardy–Littlewood–Sobolev inequality [27]).

Let $s, r > 1$ and $0 < μ < N$ with $1 / s + μ / N + 1 / r = 2$ . If $f \in L^{s} (R^{N})$ and $h \in L^{r} (R^{N})$ , then there exists a sharp constant $C (s, N, μ, r)$ , independent of f, h, such that $\begin{matrix} \int_{R^{N}} \int_{R^{N}} \frac{f (x) h (y)}{| x - y |^{μ}} ⩽ C (s, N, μ, r) | f |_{s} | h |_{r} . \end{matrix}$

The above inequality guarantees that (2.3) holds, because by $(f_{1})$ and $(f_{2})$ , given $ξ > 0$ , there is $C_{ξ} > 0$ such that $\begin{matrix} (2.4) & | F (| u |^{2}) | ⩽ ξ | u |^{2} + C_{ξ} | u |^{q} \forall u \in H_{ε} . \end{matrix}$

Then, by Hardy–Littlewood–Sobolev inequality, the integral $\begin{matrix} \int (\frac{1}{| x |^{μ}} * | F (| u |^{2}) |) | F (| u |^{2}) | \end{matrix}$ is finite if $F (| u |^{2}) \in L^{t} (R^{N}, R)$ for $t > 1$ and $\begin{matrix} \frac{2}{t} + \frac{μ}{N} = 2, \end{matrix}$ that is, $\begin{matrix} t = \frac{2 N}{2 N - μ} . \end{matrix}$ However, as $q \in (2, 2^{*} (\frac{2 N - μ}{2 N}))$ and $μ \in (0, 2)$ , (2.4) gives $\begin{matrix} \int {| F (| u |^{2}) |}^{t} < \infty \forall u \in H_{ε}, \end{matrix}$ showing (2.3).

From the above commentaries, the Euler–Lagrange functional $I_{ε} : H_{ε} \to R$ associated to ( D ε ) given by $\begin{matrix} I_{ε} (u) : = \frac{1}{2} \int | \nabla_{ε} u |^{2} + \frac{1}{2} \int V_{ε} (x) | u |^{2} - \frac{1}{4} \int (\frac{1}{| x |^{μ}} * F (| u |^{2})) F (| u |^{2}) \end{matrix}$ is well defined. Furthermore, $I_{ε} \in C^{1} (H_{ε}, R)$ with $\begin{matrix} I_{ε}^{'} (u) v = Re (\int \nabla_{ε} u \overline{\nabla_{ε} v} + V_{ε} (x) u \overline{v} - (\frac{1}{| x |^{μ}} * F (| u |^{2})) f (| u |^{2}) u \overline{v}) \forall u, v \in H_{ε} . \end{matrix}$ Hence, the weak solutions of ( D ε ) are precisely the critical points of $I_{ε}$ .

In the sequel, we intend to apply critical point theory to the functional $I_{ε}$ . However, since the problem is defined in whole space $R^{N}$ , we know that the usual Sobolev embeddings are not compact, and so, we have serious difficulties to prove that $I_{ε}$ verifies the Palais–Smale condition. In order to overcome these difficulties, we make a slightly adaptation of the penalization method introduced by del Pino and Felmer in [19] (see also [3]).

Next, we fix k large enough, which will be determined later on, and $a > 0$ to be the unique number satisfying $f (a) / a = V_{0} / k$ , with $V_{0}$ given by $(V_{1})$ . Moreover, we set $\begin{matrix} \hat{f} (s) : = \{\begin{matrix} f (s) & if s ⩽ a, \\ \frac{V_{0}}{k} & if s > a . \end{matrix} \end{matrix}$ Let $0 < t_{a} < a < T_{a}$ and consider $ϑ \in C_{0}^{\infty} (R, R)$ such that

$ϑ (s) ⩽ \hat{f} (s)$ for all $s \in [t_{a}, T_{a}]$ ;

$ϑ (t_{a}) = \hat{f} (t_{a})$ , $ϑ (T_{a}) = \hat{f} (T_{a})$ , $ϑ^{'} (t_{a}) = {\hat{f}}^{'} (t_{a})$ and $ϑ^{'} (T_{a}) = {\hat{f}}^{'} (T_{a})$ ;

the map $s \mapsto ϑ (s)$ is nondecreasing for all $s \in [t_{a}, T_{a}]$ .

By using the above functions, we define $\tilde{f} \in C^{1} (R, R)$ as follows $\begin{matrix} \tilde{f} (s) : = \{\begin{matrix} \hat{f} (s) & if s \notin [t_{a}, T_{a}], \\ ϑ (s) & if s \in [t_{a}, T_{a}] . \end{matrix} \end{matrix}$ If $χ_{Ω}$ denotes the characteristic function of the set Ω, we introduce the penalized nonlinearity $g : R^{N} \times R \to R$ by setting $\begin{matrix} (2.5) & g (x, s) : = χ_{Ω} (x) f (s) + (1 - χ_{Ω} (x)) \tilde{f} (s) . \end{matrix}$

Now, we are ready to state the modified problem $\begin{matrix} {({\tilde{D}}_{ε})}_{a} & \{\begin{matrix} {(\frac{1}{i} \nabla - A_{ε} (x))}^{2} u + V_{ε} (x) u = (\frac{1}{| x |^{μ}} * G_{ε} (x, | u |^{2})) g_{ε} (x, | u |^{2}) u, x \in R^{N}, \\ u \in H_{ε}, \end{matrix} \end{matrix}$ where $g_{ε} (x, u) : = g (ε x, u)$ and $G_{ε} (x, s) = \int_{0}^{s} g_{ε} (x, t) d t$ .

Notice that, if $\begin{matrix} Ω_{ε} : = {x \in R^{N} : ε x \in Ω} \end{matrix}$ and u is a solution of the above problem verifying $| u_{ε} (x) | ⩽ t_{a}$ in $Ω_{ε}^{c}$ , we have that $\begin{matrix} g (ε x, | u |^{2}) u = f (| u |^{2}) u for each x \in R^{N} . \end{matrix}$ Therefore, the function u is also a solution of the original problem ( D ε ).

In view of the above comment, we will deal with the existence of solutions for ( D ˜ ε ) a . Once we have $(f_{1})$ – $(f_{3})$ and $(ϑ_{1})$ – $(ϑ_{3})$ , it is possible to show that $g (x, s)$ is a Carathéodory function satisfying the following properties uniformly in $x \in R^{N}$ :

$g (x, s) = 0$ for each $s ⩽ 0$ and ${lim}_{s \to 0^{+}} g (x, s) = 0$ ;

${lim}_{s \to \infty} \frac{g (x, s)}{s^{(q - 2) / 2}} = 0$ ;

$0 ⩽ G (x, s) < g (x, s) s$ for each $x \in Ω$ , $s > 0$ ,

$0 < G (x, s) ⩽ g (x, s) s ⩽ \frac{V_{0}}{k} s$ for each $x \in Ω^{c}$ , $s > 0$ ;

moreover, we also have $\begin{matrix} s \to g (x, s) and \frac{G (x, s)}{s} are both increasing for all x \in R^{N}, s > 0 . \end{matrix}$

By $(g_{1})$ and $(g_{2})$ , the functional associated to ( D ˜ ε ) a , namely $\begin{matrix} J_{ε} (u) : = \frac{1}{2} \int | \nabla_{ε} u |^{2} + \frac{1}{2} \int V_{ε} (x) | u |^{2} - \frac{1}{4} \int (\frac{1}{| x |^{μ}} * G_{ε} (x, | u |^{2})) G_{ε} (x, | u |^{2}), u \in H_{ε} \end{matrix}$ belongs to $C^{1} (H_{ε}, R)$ . Moreover, its critical points are the weak solutions of the modified problem ( D ˜ ε ) a .

In order to show that a Palais–Smale sequence for the functional $J_{ε}$ is bounded, we observe that the lemma below holds:

Lemma 2.2.
Assume that the nonlinearity satisfies $(f_{3})$ then, for all $u \in H_{ε} ∖ {0}$ there holds $\begin{matrix} \int (\frac{1}{| x |^{μ}} * G_{ε} (x, | u |^{2})) g_{ε} (x, | u |^{2}) | u |^{2} - \int (\frac{1}{| x |^{μ}} * G_{ε} (x, | u |^{2})) G_{ε} (x, | u |^{2}) ⩾ 0 . \end{matrix}$
Proof.
The above inequality follows from $(g_{3})$ . □
Lemma 2.3.
The penalized functional $J_{ε}$ has a ${(PS)}_{m_{ε}}$ sequence $(u_{n}) \subset H_{ε}$ , where the minimax value $m_{ε}$ is defined by $\begin{matrix} (2.6) & m_{ε} : = inf_{u \in E ∖ {0}} max_{t ⩾ 0} J_{ε} (t u) = inf_{u \in N_{ε}} J_{ε} (u), \end{matrix}$ that is, $\begin{matrix} J_{ε}^{'} (u_{n}) \to 0 and J_{ε} (u_{n}) \to m_{ε} . \end{matrix}$ Moreover, there exists a constant $d > 0$ , independent of ε, k, a, such that $m_{ε} < d$ for all ε small.
Proof.
The lemma follows showing that $J_{ε}$ satisfies the Mountain Pass geometry:
There exist $ρ, δ_{0} > 0$ such that $J_{ε} |_{S} ⩾ δ_{0} > 0$ for all $u \in S = {u \in E : ∥ u ∥ = ρ}$ .

From $(g_{1})$ – $(g_{2})$ and the Hardy–Littlewood–Sobolev inequality, we derive that $\begin{matrix} \begin{matrix} J_{ε} (u) ⩾ \frac{1}{2} ∥ u ∥_{ε}^{2} - C (∥ u ∥_{ε}^{4} + ∥ u ∥_{ε}^{2 q}) . \end{matrix} \end{matrix}$ The claim follows if we choose ρ small enough.
There is $r > 0$ and e with $∥ e ∥ > r$ such that $J_{ε} (e) < 0$ .

Take a nonnegative function $u_{0} \in H_{ε} ∖ {0}$ with $Supp (u_{0}) \subset Ω_{ε}$ and observe that $\begin{matrix} G (ε x, u_{0}) = F (u_{0}) . \end{matrix}$ Set $\begin{matrix} g (t) = F (\frac{t u_{0}}{∥ u_{0} ∥}) > 0 for t > 0, \end{matrix}$ where $\begin{matrix} (2.7) & F (u) = \frac{1}{4} \int_{R^{N}} (\frac{1}{| x |^{μ}} * F (| u |^{2})) F (| u |^{2}) . \end{matrix}$ By condition $(f_{2})$ , $\begin{matrix} F (t) ⩽ f (t) t, t > 0, \end{matrix}$ then $\begin{matrix} \frac{g^{'} (t)}{g (t)} ⩾ \frac{4}{t} for all t > 0 . \end{matrix}$ Integrating this over $[1, s ∥ u_{0} ∥]$ with $s > \frac{1}{∥ u_{0} ∥}$ we find $\begin{matrix} F (s u_{0}) ⩾ F (\frac{u_{0}}{∥ u_{0} ∥}) ∥ u_{0} ∥^{4} s^{4} . \end{matrix}$ Therefore $\begin{matrix} J_{ε} (s u_{0}) ⩽ C_{1} s^{2} - C_{2} s^{4} for s > \frac{1}{∥ u_{0} ∥}, \end{matrix}$ and (2) holds for $e = s u_{0}$ and s large enough.

Using the Mountain Pass theorem without (PS) condition [41], we obtain the existence of a ${(PS)}_{m_{ε}}$ sequence $(u_{n}) \subset H_{ε}$ with $\begin{matrix} m_{ε} : = inf_{u \in E ∖ {0}} max_{t ⩾ 0} J_{ε} (t u) = inf_{u \in N_{ε}} J_{ε} (u) . \end{matrix}$ Since $Supp (u_{0}) \subset Ω_{ε}$ , it is easy to obtain the existence of constant $d > 0$ , independent of ε, ℓ, a, such that $m_{ε} < d$ for all ε small. □
Lemma 2.4.
If $(u_{n}) \subset H_{ε}$ is a ${(PS)}_{m_{ε}}$ sequence for $J_{ε}$ , then $(u_{n})$ is bounded and there is $n_{0} \in N$ such that $∥ u_{n} ∥_{ε}^{2} ⩽ 4 (d + 1)$ for all $n ⩾ n_{0}$ .
Proof.
As $J_{ε} (u_{n}) \to m_{ε}$ and $J_{ε}^{'} (u_{n}) \to 0$ , $(g_{3})$ and Lemma 2.2 combine to give $\begin{array}{rcl} m_{ε} + o_{n} (1) ∥ u_{n} ∥_{ε} & ⩾ & J_{ε} (u_{n}) - \frac{1}{4} J_{ε}^{'} (u_{n}) u_{n} \\ ⩾ & (\frac{1}{2} - \frac{1}{4}) ∥ u_{n} ∥_{ε}^{2}, \end{array}$ where $o_{n} (1)$ denotes a quantity approaching zero as $n \to \infty$ . From this, $(u_{n})$ is bounded in $H_{ε}$ and $∥ u_{n} ∥_{ε}^{2} ⩽ 4 (d + 1)$ for n large. □

Before proving the next lemma, we need to fix some notations. In what follows, $\begin{matrix} (2.8) & B : = {u \in H_{ε} : ∥ u_{n} ∥_{ε}^{2} ⩽ 4 (d + 1)} \end{matrix}$ and $\begin{matrix} K (u) (x) : = \frac{1}{| x |^{μ}} * G (x, | u |^{2}) . \end{matrix}$ With the above notations, we are able to show the following estimate.
Lemma 2.5.
Suppose that $(f_{1})$ – $(f_{3})$ hold. Then, there exists $k_{0}$ such that $\begin{matrix} \frac{{sup}_{u \in B} | K (u) (x) |_{L^{\infty} (R^{N})}}{k_{0}} < \frac{1}{2} . \end{matrix}$
Proof.
Notice that $\begin{matrix} | G (x, s) | ⩽ | F (s) | \forall s \in R . \end{matrix}$ Thereby, $\begin{matrix} | K (u) (x) | ⩽ | \int \frac{F (| u |^{2})}{| x - y |^{μ}} d y | ⩽ | \int_{| x - y | ⩽ 1} \frac{F (| u |^{2})}{| x - y |^{μ}} d y | + | \int_{| x - y | ⩾ 1} \frac{F (| u |^{2})}{| x - y |^{μ}} d y | . \end{matrix}$ Using $(f_{1})$ and $(f_{2})$ , given $ξ > 0$ there exists $c_{ξ} > 0$ such that $\begin{array}{rcl} | K (u) (x) | & ⩽ & \frac{ξ}{2} \int_{| x - y | ⩽ 1} \frac{| u |^{2}}{| x - y |^{μ}} d y + \frac{C_{ξ}}{q} \int_{| x - y | ⩽ 1} \frac{| u |^{q}}{| x - y |^{μ}} d y \\ + \frac{ξ}{2} \int_{| x - y | ⩾ 1} \frac{| u |^{2}}{| x - y |^{μ}} d y + \frac{C_{ξ}}{q} \int_{| x - y | ⩾ 1} \frac{| u |^{q}}{| x - y |^{μ}} d y . \end{array}$ From Sobolev’s embedding, there is $C_{1} > 0$ such that $\begin{matrix} | K (u) (x) | ⩽ \frac{ξ}{2} \int_{| x - y | ⩽ 1} \frac{| u |^{2}}{| x - y |^{μ}} d y + \frac{C_{ξ}}{q} \int_{| x - y | ⩽ 1} \frac{| u |^{q}}{| x - y |^{μ}} d y + C_{1} . \end{matrix}$ Once $μ \in (0, 2)$ and $q \in (2, \frac{2 (N - μ)}{N - 2})$ , we fix $t \in (\frac{N}{N - μ}, \frac{N}{N - 2}]$ and $s \in (\frac{N}{N - μ}, \frac{2 N}{(N - 2) q}]$ . Then, by Hölder inequality, $\begin{matrix} \begin{matrix} \int_{| x - y | ⩽ 1} \frac{| v |^{2}}{| x - y |^{μ}} & ⩽ {(\int_{| x - y | ⩽ 1} | v |^{2 t})}^{\frac{1}{t}} {(\int_{| x - y | ⩽ 1} \frac{1}{| x - y |^{\frac{t μ}{t - 1}}})}^{\frac{t - 1}{t}} \\ ⩽ C_{2} {(\int_{| r | ⩽ 1} | r |^{N - 1 - \frac{t μ}{t - 1}} d r)}^{\frac{t - 1}{t}} \end{matrix} \end{matrix}$ and $\begin{matrix} \begin{matrix} \int_{| x - y | ⩽ 1} \frac{| v |^{q}}{| x - y |^{μ}} & ⩽ {(\int_{| x - y | ⩽ 1} | v |^{s q})}^{\frac{1}{s}} {(\int_{| x - y | ⩽ 1} \frac{1}{| x - y |^{\frac{s μ}{s - 1}}})}^{\frac{s - 1}{s}} \\ ⩽ C_{2} {(\int_{| r | ⩽ 1} | r |^{N - 1 - \frac{s μ}{s - 1}} d r)}^{\frac{s - 1}{s}} . \end{matrix} \end{matrix}$ As $N - 1 - \frac{t μ}{t - 1} > - 1$ and $N - 1 - \frac{s μ}{s - 1} > - 1$ , there is $C_{3} > 0$ such that $\begin{matrix} \int_{| x - y | ⩽ 1} \frac{(| v |^{2} + | v |^{q})}{| x - y |^{μ}} ⩽ C_{3} \forall x \in R^{N} . \end{matrix}$ Thus there exists $k_{0} > 0$ such that $\begin{matrix} (2.9) & \frac{{sup}_{u \in B} | K (u) (x) |_{L^{\infty} (R^{N})}}{k_{0}} ⩽ \frac{1}{2} . □ \end{matrix}$

From now on, we take $k > k_{0}$ and consider the penalized problem with nonlinearity defined in (2.5).
Lemma 2.6.
Let $(u_{n}) \subset H_{ε}$ be a ${(PS)}_{m_{ε}}$ sequence for $J_{ε}$ . Then, for each $ξ > 0$ there exists $R = R (ξ) > 0$ such that $Ω_{ε} \subset B_{R} (0)$ and $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{B_{R} {(0)}^{c}} (| \nabla_{ε} u_{n} |^{2} + V_{ε} (x) | u_{n} |^{2}) ⩽ ξ . \end{matrix}$
Proof.
From Lemma 2.4, $\begin{matrix} ∥ u_{n} ∥_{ε}^{2} ⩽ 4 (d + 1) \forall n \in N, \end{matrix}$ where d is independent of the choice of $k_{0}$ . From this, we can suppose that there exists $u \in H_{ε}$ such that $u_{n} ⇀ u$ in $H_{ε}$ .

Next, we consider $η_{R} \in C^{\infty} (R^{N}, R)$ such that $0 ⩽ η_{R} ⩽ 1$ , $η_{R} \equiv 0$ in $B_{R / 2} (0)$ , $η_{R} \equiv 1$ in $B_{R} {(0)}^{c}$ and $| \nabla η_{R} | ⩽ C / R$ , where $C > 0$ is a constant independent of R. Once the sequence $(η_{R} u_{n})$ is bounded in $H_{ε}$ , we have that $J_{ε}^{'} (u_{n}) (η_{R} u_{n}) = o_{n} (1)$ , that is, $\begin{array}{rcl} Re (\int \nabla_{ε} u_{n} \overline{\nabla_{ε} (u_{n} η_{R})}) + \int V_{ε} (x) | u_{n} |^{2} η_{R} \\ = \int (\frac{1}{| x |^{μ}} * G_{ε} (x, | u_{n} |^{2})) g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} η_{R} + o_{n} (1) . \end{array}$

On the other hand, as $η_{R}$ takes values in $R$ , a direct calculation shows that $\begin{matrix} \overline{\nabla_{ε} (u_{n} η_{R})} = i {\overline{u}}_{n} \nabla η_{R} + η_{R} \overline{\nabla_{ε} u_{n}} . \end{matrix}$ The two above equalities and $(g_{3})$ (ii) imply that $\begin{matrix} \begin{matrix} \int (| \nabla_{ε} u_{n} |^{2} + V_{ε} (x) | u_{n} |^{2}) η_{R} ⩽ & \int \frac{{sup}_{u \in B} | K (u) (x) |_{L^{\infty} (R^{N})}}{k_{0}} V_{ε} (x) | u_{n} |^{2} η_{R} \\ + Re (\int - i \overline{u_{n}} \nabla_{ε} u_{n} \nabla η_{R}) + o_{n} (1) . \end{matrix} \end{matrix}$ Now, using the definition of $η_{R}$ together with the Lemma 2.5, the Hölder’s inequality and the boundedness of $(u_{n})$ , we obtain $\begin{matrix} \begin{matrix} \frac{1}{2} \int_{B_{R} {(0)}^{c}} (| \nabla_{ε} u_{n} |^{2} + V_{ε} (x) | u_{n} |^{2}) & ⩽ \frac{C}{R} ∥ \overline{u_{n}} ∥_{L^{2}} ∥ \nabla_{ε} u_{n} ∥_{L^{2}} + o_{n} (1) \\ ⩽ \frac{C_{1}}{R} + o_{n} (1) . \end{matrix} \end{matrix}$ So, for any fixed $ξ > 0$ , we can choose $R > 0$ large enough in such way that $Ω_{ε} \subset B_{R} (0)$ and $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{B_{R} {(0)}^{c}} (| \nabla_{ε} u |^{2} + V_{ε} (x) | u_{n} |^{2}) ⩽ ξ . \end{matrix}$ This finishes the proof. □

The lemma below shows that the modified functional satisfies the Palais–Smale condition at some levels.
Lemma 2.7.
The functional $J_{ε}$ satisfies the ${(PS)}_{c}$ condition for any $c \in [0, d]$ .
Proof.
Since $u_{n} ⇀ u$ in $H_{ε}$ and $J_{ε}^{'} (u_{n}) u_{n} = J_{ε}^{'} (u_{n}) u = o_{n} (1)$ , it follows that $\begin{matrix} ∥ u_{n} ∥_{ε}^{2} - ∥ u_{n} ∥_{ε}^{2} = ∥ u_{n} - u ∥_{ε}^{2} + o_{n} (1) = \int_{R^{N}} K (u_{n}) g_{ε} (x, | u_{n} |^{2}) (| u_{n} |^{2} - | u |^{2}) + o_{n} (1) . \end{matrix}$ In what follows, our goal is to show $\begin{matrix} \int K (u_{n}) g_{ε} (x, | u_{n} |^{2}) (| u_{n} |^{2} - | u |^{2}) = o_{n} (1) . \end{matrix}$ From Lemma 2.5, we know that there exists $C > 0$ such that $\begin{matrix} | K (u_{n}) | ⩽ C \forall n \in N . \end{matrix}$ Thereby, for any fixed $R > 0$ , the Sobolev’s compact embedding combined with subcritical growth of g yields $\begin{matrix} | \int_{B_{R}} K (u_{n}) g_{ε} (x, | u_{n} |^{2}) (| u_{n} |^{2} - | u |^{2}) | ⩽ C \int_{B_{R}} | g_{ε} (x, | u_{n} |^{2}) (| u_{n} |^{2} - | u |^{2}) | \to 0 . \end{matrix}$ From Lemma 2.6, we see that for each $ξ > 0$ , there exists $R = R (ξ) > 0$ such that $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{R^{N} ∖ B_{R}} K (u_{n}) | g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} | ⩽ C_{1} ξ . \end{matrix}$ Similarly, by Hölder inequality, $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{R^{N} ∖ B_{R}} K (u_{n}) | g_{ε} (x, | u_{n} |^{2}) | u |^{2} | ⩽ C_{2} ξ . \end{matrix}$ In conclusion, there holds $\begin{matrix} \int K (u_{n}) g_{ε} (x, | u_{n} |^{2}) (| u_{n} |^{2} - | u |^{2}) \to 0 . □ \end{matrix}$

Let us denote by $N_{ε}$ the Nehari manifold of $J_{ε}$ , namely $\begin{matrix} N_{ε} : = {u \in H_{ε} ∖ {0} : J_{ε}^{'} (u) u = 0} . \end{matrix}$ Gathering the growth condition of g, Lemma 2.5 and Proposition 2.1, we can show that there exists $r > 0$ , independent of $ε > 0$ , such that $\begin{matrix} (2.10) & ∥ u ∥_{ε} ⩾ r > 0 for each u \in N_{ε} . \end{matrix}$

Next, we prove the main result of this section, which is a key point to show the multiplicity of solution for $({\tilde{D}}_{ε})$ . Proposition 2.8.
The functional $J_{ε}$ restricted to $N_{ε}$ satisfies the ${(PS)}_{c}$ condition for any $c \in [0, d]$ .
Proof.
Let $(u_{n}) \subset N_{ε}$ be a ${(PS)}_{c}$ sequence of $J_{ε}$ restricted to $N_{ε}$ . Then, there exists $(λ_{n}) \subset R$ such that $\begin{matrix} (2.11) & J_{ε}^{'} (u_{n}) = λ_{n} ϕ_{ε}^{'} (u_{n}) + o_{n} (1), \end{matrix}$ where $ϕ_{ε} : H_{ε} \to R$ is given by $\begin{matrix} ϕ_{ε} (u) : = J_{ε}^{'} (u) u = \int (| \nabla_{ε} u |^{2} + V_{ε} (x) | u |^{2} - K (u) g_{ε} (x, | u |^{2}) | u |^{2}) . \end{matrix}$

For any fixed $u \in N_{ε}$ , the function $g (ε x, | u (x) |^{2})$ is constant on $Ω_{ε}^{c} \cap {| u |^{2} > T_{a}}$ . Hence, gathering the definition of g, $(f_{4})$ and the monotonicity of ϑ, we find $\begin{array}{rcl} ϕ_{ε}^{'} (u) u & = & 2 ∥ u ∥_{ε}^{2} - 2 \int (\frac{1}{| x |^{μ}} * g_{ε} (x, | u |^{2}) (| u |^{2})) g_{ε} (x, | u |^{2}) | u |^{2} \\ - 2 \int (\frac{1}{| x |^{μ}} * G_{ε} (x, | u |^{2})) g_{ε}^{'} (x, | u |^{2}) | u |^{4} \\ - 2 \int (\frac{1}{| x |^{μ}} * G_{ε} (x, | u |^{2})) g_{ε} (x, | u |^{2}) | u |^{2} \\ ⩽ & - 2 \int (\frac{1}{| x |^{μ}} * G_{ε} (x, | u |^{2})) g_{ε} (x, | u |^{2}) | u |^{2} . \end{array}$

By the boundedness of $(u_{n})$ and Lemma 2.5, we can assume that $ϕ_{ε}^{'} (u_{n}) u_{n} \to l ⩽ 0$ . If $l \neq 0$ , we infer from (2.11) that $λ_{n} = o_{n} (1)$ . In this case, using (2.11) again, we conclude that $(u_{n})$ is a ${(PS)}_{d}$ sequence for the unconstrained functional. Consequently, by Proposition 2.7, we find a strongly convergent subsequence in $H_{ε}$ .

It remains to prove that $l \neq 0$ . Suppose, by contradiction, that $l = 0$ . As $Ω_{ε}$ is a bounded domain, by Sobolev imbedding, $u_{n} \to 0$ in $L^{q} (Ω_{ε})$ . Hence, we can use the equality $J_{ε}^{'} (u_{n}) u_{n} = 0$ , the subcritical growth of g, the Lemma 2.5 and $(g_{4})$ (ii) to conclude that $\begin{matrix} ∥ u_{n} ∥_{ε}^{2} = \int_{Ω_{ε}^{c}} K (u_{n}) g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} + o_{n} (1) ⩽ \frac{1}{2} \int_{Ω_{ε}^{c}} V_{ε} (x) | u_{n} |^{2} + o_{n} (1) . \end{matrix}$ The above expression implies that $∥ u_{n} ∥_{ε}^{2} \to 0$ , which leads to a contradiction with (2.10). This contradiction concludes the proof. □

3. Multiple solutions for the modified problem

In this section, we will study the multiplicity of solutions for ( D ˜ ε ) a . In the sequel, we assume that $δ > 0$ is small enough in such way that $M_{δ} \subset Ω$ .

We start by considering the limiting problem associated to ( D ε ), namely the scalar problem $\begin{matrix} (A_{0}) & \{\begin{matrix} - Δ v + V_{0} v = (\frac{1}{| x |^{μ}} * F (| v |^{2})) f (| v |^{2}) v in R^{N}, \\ v \in H^{1} (R^{N}, R), \end{matrix} \end{matrix}$ which has the following energy functional $\begin{matrix} E_{0} (v) : = \frac{1}{2} \int | \nabla v |^{2} + \frac{1}{2} \int V_{0} | v |^{2} - \frac{1}{4} \int (\frac{1}{| x |^{μ}} * F (| v |^{2})) F (| v |^{2}), v \in H^{1} (R^{N}, R) . \end{matrix}$ We also consider $\begin{matrix} M_{0} : = {v \in H^{1} (R^{N}, R) ∖ {0} : E_{0}^{'} (v) v = 0} \end{matrix}$ and the level $\begin{matrix} (3.1) & c_{0} = inf_{v \in M_{0}} E_{0} (v) = inf_{v \in H^{1} (R^{N}, R) ∖ {0}} max_{t ⩾ 0} E_{0} (t v) > 0 . \end{matrix}$

Using the invariance of $E_{0}$ by translations and applying a result due to Lions [31, Lemma I.1], it follows that ( A 0 ) has a nontrivial ground state solution, that is, a positive function $ω \in H^{1} (R^{N}, R)$ such that $E_{0} (ω) = c_{0}$ and $E_{0}^{'} (ω) = 0$ . From now on, we will denote by ω such solution.

The lemma below is an important step to prove the concentration of the solutions that we will find in the next section. Once the it follows with the same arguments explored [2, Theorem 3.1], we will omit its proof.

Lemma 3.1.
Let $(w_{n}) \subset M_{0}$ be such that $E_{0} (w_{n}) \to c_{0}$ . Then, for some subsequence of $(w_{n})$ , still denoted by itself, there exists a sequence $({\tilde{y}}_{n}) \subset R^{N}$ such that $w_{n} (\cdot + {\tilde{y}}_{n}) \to w_{1} \in M_{0}$ with $E_{0} (w_{1}) = c_{0}$ .

Let $ψ \in C^{\infty} (R^{+}, [0, 1])$ be such that $ψ \equiv 1$ in $[0, δ / 2]$ and $ψ \equiv 0$ in $[δ, \infty)$ . We define for each $y \in M$ the function $\begin{matrix} Ψ_{ε, y} (x) : = ψ (| ε x - y |) ω (\frac{ε x - y}{ε}) exp (i τ_{y} (\frac{ε x - y}{ε})), \end{matrix}$ where $τ_{y} (x) : = \sum_{j = 1}^{N} A_{j} (x) x_{j}$ . Let $t_{ε} > 0$ be the unique positive number such that $\begin{matrix} max_{t ⩾ 0} J_{ε} (t Ψ_{ε, y}) = J_{ε} (t_{ε} Ψ_{ε, y}) . \end{matrix}$ By noticing that $t_{ε} Ψ_{ε, y} \in N_{ε}$ , we can consider the function $Φ_{ε} : M \to N_{ε}$ given by $\begin{matrix} Φ_{ε} (y) : = t_{ε} Ψ_{ε, y} . \end{matrix}$

The energy of the above function has the following behavior as ε goes to 0.
Lemma 3.2.
The family $(Φ_{ε})$ verifies the limit below $\begin{matrix} lim_{ε \to 0^{+}} J_{ε} (Φ_{ε} (y)) = c_{0}, \end{matrix}$ uniformly for $y \in M$ .
Proof.
Arguing by contradiction, we suppose that there exist $γ > 0$ , $(y_{n}) \subset M$ and $ε_{n} \to 0^{+}$ verifying $\begin{matrix} (3.2) & | J_{ε_{n}} (Φ_{ε_{n}} (y_{n})) - c_{0} | ⩾ γ > 0 \forall n \in N . \end{matrix}$ In order to simplify the notation, we write $Φ_{n}$ , $Ψ_{n}$ and $t_{n}$ to denote $Φ_{ε_{n}} (y_{n})$ , $Ψ_{ε_{n}, y_{n}}$ and $t_{ε_{n}}$ , respectively.

Arguing as in [12, Lemma 3.2], we see that $\begin{matrix} (3.3) & ∥ Ψ_{n} ∥_{ε_{n}}^{2} \to \int (| \nabla ω |^{2} + V_{0} | ω |^{2}) . \end{matrix}$ On the other hand, as $J_{ε_{n}}^{'} (t_{n} Ψ_{n}) (t_{n} Ψ_{n}) = 0$ , the change of variables $z : = (ε_{n} x - y_{n}) / ε_{n}$ provides $\begin{matrix} \begin{matrix} ∥ t_{n} Ψ_{n} ∥_{ε_{n}}^{2} & = \int K (t_{n} Ψ_{n}) g (ε_{n} x, {| t_{n} Ψ_{n} (x) |}^{2}) {| t_{n} Ψ_{n} (x) |}^{2} d x \\ = \int K (t_{n} Ψ_{n}) g (ε_{n} z + y_{n}, {| t_{n} ψ (| ε_{n} z |) ω (z) |}^{2}) {| t_{n} ψ (| ε_{n} z |) ω (z) |}^{2} d z . \end{matrix} \end{matrix}$

If $z \in B_{δ / ε_{n}} (0)$ , then $ε_{n} z + y_{n} \in B_{δ} (y_{n}) \subset M_{δ} \subset Ω$ . Thus, as $g (x, s) = f (s)$ for any $x \in Ω$ and $ψ (s) = 0$ for $s ⩾ δ$ , the last equality leads to $\begin{matrix} (3.4) & ∥ Ψ_{n} ∥_{ε_{n}}^{2} = \int K (t_{n} ψ (| ε_{n} z |) ω (z)) f ({| t_{n} ψ (| ε_{n} z |) ω (z) |}^{2}) {| ψ (| ε_{n} z |) ω (z) |}^{2} d z . \end{matrix}$

Let $α : = min {w (z) : | z | ⩽ δ / 2}$ and $n_{0} \in N$ is such that $B_{δ / 2} (0) \subset B_{δ / (2 ε_{n})} (0)$ for all $n ⩾ n_{0}$ . Then, $\begin{matrix} (3.5) & ∥ Ψ_{n} ∥_{ε_{n}}^{2} ⩾ F (| t_{n} α |^{2}) f (| t_{n} α |^{2}) \int_{B_{δ / 2} (0)} \int_{B_{δ / 2} (0)} \frac{| ω (y) |^{2} | ω (z) |^{2}}{| y - z |^{μ}} d y d z \end{matrix}$ for all $n ⩾ n_{0}$ , where we have used that F and f are increasing.

If $| t_{n} | \to \infty$ , (3.5) combined with $(f_{3})$ gives $∥ Ψ_{n} ∥_{ε_{n}}^{2} \to + \infty$ , contradicting (3.3). Thus, up to a subsequence, $t_{n} \to t_{0} ⩾ 0$ .

Since g has subcritical growth and $t_{n} Ψ_{n} \in N_{ε_{n}}$ , it follows that $t_{0} > 0$ . Thereby, taking the limit in (3.4), we obtain $\begin{matrix} \int {| \nabla (t_{0} ω) |}^{2} + {| (t_{0} ω) |}^{2} = \int K (t_{0} ω) f (| t_{0} ω |^{2}) | t_{0} ω |^{2}, \end{matrix}$ from which follows that $t_{0} ω \in M_{0}$ . Once ω also belongs to $M_{0}$ , we deduce that $t_{0} = 1$ . This fact together with Lebesgue’s theorem imply that $\begin{matrix} \int K (t_{n} Ψ_{n}) F (| t_{n} Ψ_{n} |^{2}) \to \int K (ω) F (ω) | ω |^{2} . \end{matrix}$ Hence, $\begin{matrix} lim_{n \to \infty} J_{ε_{n}} (Φ_{ε_{n}} (y_{n})) = E_{0} (ω) = c_{0}, \end{matrix}$ which contradicts (3.2), finishing the proof of the lemma. □

Let us fix $ρ = ρ_{δ} > 0$ in such way that $M_{δ} \subset B_{ρ} (0)$ and define $Υ : R^{N} \to R^{N}$ by setting $Υ (x) : = x$ for $| x | < ρ$ and $Υ (x) : = ρ x / | x |$ for $| x | ⩾ ρ$ . We also consider the barycenter map $β_{ε} : N_{ε} \to R^{N}$ given by $\begin{matrix} β_{ε} (u) : = \frac{\int Υ (ε x) | u (x) |^{2}}{\int | u (x) |^{2}} . \end{matrix}$ Here, we would like point out that by a direct computation, we derive that $\begin{matrix} (3.6) & lim_{ε \to 0} β_{ε} (Φ_{ε} (y)) = y uniformly for y \in M . \end{matrix}$ The above limit is crucial to apply some results involving category of sets, see proof of Theorem 3.6 later on.

In what follows, we consider the following subset of the Nehari manifold $\begin{matrix} (3.7) & {\tilde{N}}_{ε} : = {u \in N_{ε} : J_{ε} (u) ⩽ c_{0} + h (ε)}, \end{matrix}$ where $h : R^{+} \to R^{+}$ is given by $\begin{matrix} h (ε) = sup_{y \in M} | J_{ε} (Φ_{ε} (y)) - c_{0} | . \end{matrix}$ Thus, $Φ_{ε} (y) \in {\tilde{N}}_{ε}$ for all $y \in M$ , showing that ${\tilde{N}}_{ε} \neq \emptyset$ for any $ε > 0$ . In what follows, we will use the limit below $\begin{matrix} h (ε) \to 0 as ε \to 0^{+}, \end{matrix}$ which is an immediate consequence of Lemma 3.2.

We present below an interesting relation between ${\tilde{N}}_{ε}$ and the barycenter map.
Proposition 3.3.
For any $δ > 0$ we have that $\begin{matrix} lim_{ε \to 0^{+}} sup_{u \in {\tilde{N}}_{ε}} dist (β_{ε} (u), M_{δ}) = 0 . \end{matrix}$

The above result is a version of [12, Lemma 4.1]. Since we are working with nonhomogeneous nonlinearities, the arguments used there do not work well in our problem. Consequently, we need of other approach. We are going to use the following compactness result.
Lemma 3.4.
Let $ε_{n} \to 0^{+}$ and $(u_{n}) \subset N_{ε_{n}}$ be such that $J_{ε_{n}} (u_{n}) \to c_{0}$ . Then there exists a sequence $({\tilde{y}}_{n}) \subset R^{N}$ such that $v_{n} : = | u_{n} | (\cdot + {\tilde{y}}_{n})$ has a convergent subsequence in $H^{1} (R^{N}, R)$ . Moreover, up to a subsequence, $ε_{n} {\tilde{y}}_{n} \to y_{0} \in M$ .

By assuming the above result, we are ready to prove Proposition 3.3 as follows.
Proof of Proposition 3.3.
Let $(ε_{n}) \subset R$ be such that $ε_{n} \to 0^{+}$ . By definition, there exists $(u_{n}) \subset {\tilde{N}}_{ε_{n}}$ such that $\begin{matrix} dist (β_{ε_{n}} (u_{n}), M_{δ}) = sup_{u \in {\tilde{N}}_{ε_{n}}} dist (β_{ε_{n}} (u), M_{δ}) + o_{n} (1) . \end{matrix}$ Thus, it suffices to find a sequence $(y_{n}) \subset M_{δ}$ such that $\begin{matrix} (3.8) & | β_{ε_{n}} (u_{n}) - y_{n} | = o_{n} (1) . \end{matrix}$

It follows from the diamagnetic inequality (2.1) that $E_{0} (t u_{n}) ⩽ J_{ε_{n}} (t u_{n})$ . Therefore, recalling that $(u_{n}) \subset {\tilde{N}}_{ε_{n}} \subset N_{ε_{n}}$ , we see that (3.1) loads to $\begin{matrix} c_{0} ⩽ max_{t ⩾ 0} E_{0} (t u_{n}) ⩽ max_{t ⩾ 0} J_{ε_{n}} (t u_{n}) = J_{ε_{n}} (u_{n}) ⩽ c_{0} + h (ε_{n}), \end{matrix}$ from which follows that $J_{ε_{n}} (u_{n}) \to c_{0}$ . Thus, we can invoke Lemma 3.4 to obtain a sequence $({\tilde{y}}_{n}) \subset R^{N}$ such that $(y_{n}) : = (ε_{n} {\tilde{y}}_{n}) \subset M_{δ}$ and $| u_{n} | (\cdot + {\tilde{y}}_{n})$ is strongly convergent in $H^{1} (R^{N}, R)$ . From this, $\begin{matrix} \begin{matrix} β_{ε_{n}} (u_{n}) & = \frac{\int Υ (ε_{n} x) | u_{n} |^{2}}{\int | u_{n} |^{2}} = \frac{\int Υ (ε_{n} z + y_{n}) | u_{n} (z + {\tilde{y}}_{n}) |^{2}}{\int | u_{n} (z + {\tilde{y}}_{n}) |^{2}} \\ = y_{n} + \frac{\int (Υ (ε_{n} z + y_{n}) - y_{n}) | u_{n} (z + {\tilde{y}}_{n}) |^{2}}{\int | u_{n} (z + {\tilde{y}}_{n}) |^{2}} = y_{n} + o_{n} (1), \end{matrix} \end{matrix}$ showing that $(y_{n})$ satisfies (3.8), finishing the proof. □

Next, we proceed with the proof of Lemma 3.4.
Proof of Lemma 3.4.
As in Lemma 2.7, $(u_{n})$ is bounded in $H_{ε}$ . We claim that there exist a sequence $({\tilde{y}}_{n}) \subset R^{N}$ and constants $R, γ > 0$ such that $\begin{matrix} (3.9) & \underset{n \to \infty}{lim inf} \int_{B_{R} ({\tilde{y}}_{n})} | u_{n} |^{2} ⩾ γ > 0 . \end{matrix}$ Indeed, if this is not true, gathering the boundedness of $(| u_{n} |)$ in $H^{1} (R^{N}, R)$ , Lemma 2.5 and a well-known lemma due to Lions [31, Lemma I.1], we can ensure that $| u_{n} | \to 0$ in $L^{s} (R^{N})$ for all $2 < s < 2^{}$ . On the other hand, given $ξ > 0$ , it follows from $(g_{2})$ and $(g_{3})$ $\begin{matrix} ∥ u_{n} ∥_{ε_{n}}^{2} = \int K (u_{n}) g (ε x, | u_{n} |^{2}) | u_{n} |^{2} ⩽ C ξ \int | u_{n} |^{2} + C C_{ξ} \int | u_{n} |^{q} . \end{matrix}$ Once $u_{n} \to 0$ in $L^{q} (R^{N})$ and ξ is arbitrary, we conclude that $∥ u_{n} ∥_{ε_{n}} \to 0$ . Therefore, $J_{ε_{n}} (u_{n}) \to 0$ , contradicting $c_{0} > 0$ . Thereby, (3.9) holds, and for some subsequence, $\begin{matrix} v_{n} : = | u_{n} | (\cdot + {\tilde{y}}_{n}) ⇀ v \neq 0 weakly in H^{1} (R^{N}, R) . \end{matrix}$

We now consider $t_{n} > 0$ such that $w_{n} : = t_{n} v_{n} \in M_{0}$ . It follows from the diamagnetic inequality (2.1) that $\begin{matrix} (3.10) & c_{0} ⩽ E_{0} (w_{n}) ⩽ max_{t ⩾ 0} J_{ε_{n}} (t v_{n}) = J_{ε_{n}} (u_{n}) = c_{0} + o_{n} (1) . \end{matrix}$ Hence $E_{0} (w_{n}) \to c_{0}$ , and so, $w_{n} ↛ 0$ in $H^{1} (R^{N}, R)$ .

Since $(v_{n})$ and $(w_{n})$ are bounded in $H^{1} (R^{N}, R)$ and $v_{n} ↛ 0$ in $H^{1} (R^{N}, R)$ , the sequence $(t_{n})$ is bounded. Thus, up to a subsequence, $t_{n} \to t_{0} ⩾ 0$ . If $t_{0} = 0$ then $∥ w_{n} ∥_{H^{1} (R^{N}, R)} \to 0$ , which does not occurs. Hence $t_{0} > 0$ , and so, the sequence $(w_{n})$ satisfies $\begin{matrix} E_{0} (w_{n}) \to c_{0}, w_{n} ⇀ w : = t_{0} v \neq 0 weakly in H^{1} (R^{N}, R) . \end{matrix}$ It follows from Lemma 3.1 that $w_{n} \to w$ , or equivalently, $v_{n} \to v$ in $H^{1} (R^{N}, R)$ . This proves the first part of the lemma.

In order to finish the proof, we set $y_{n} : = ε_{n} {\tilde{y}}_{n}$ and claim that $(y_{n})$ has a bounded subsequence. Indeed, otherwise $| y_{n} | \to \infty$ . Thereby, fixing $R > 0$ such that $Ω \subset B_{R} (0)$ , we can assume that $| y_{n} | > 2 R$ for all $n \in N$ . Therefore, for any $z \in B_{R / ε_{n}} (0)$ $\begin{matrix} | ε_{n} z + y_{n} | ⩾ | y_{n} | - | ε_{n} z | > R . \end{matrix}$ If we now set $Γ_{n} : = B_{R / ε_{n}} (0)$ , we can gather $(u_{n}) \subset N_{ε_{n}}$ , $(V_{1})$ , (2.1), the change of variables $x \mapsto z + {\tilde{y}}_{n}$ , the above expression and (2.5) to get $\begin{matrix} \begin{matrix} \int | \nabla v_{n} |^{2} + V_{0} \int | v_{n} |^{2} & ⩽ \int K (v_{n}) g (ε_{n} z + y_{n}, | v_{n} |^{2}) | v_{n} |^{2} \\ = \int_{Γ_{n}} K (v_{n}) \tilde{f} (| v_{n} |^{2}) | v_{n} |^{2} + \int_{Γ_{n}^{c}} K (v_{n}) g (ε_{n} z + y_{n}, | v_{n} |^{2}) | v_{n} |^{2} \\ ⩽ \int_{Γ_{n}} K (v_{n}) \tilde{f} (| v_{n} |^{2}) | v_{n} |^{2} + \int_{Γ_{n}^{c}} K (v_{n}) f (| v_{n} |^{2}) | v_{n} |^{2} . \end{matrix} \end{matrix}$ Since $v_{n} \to v$ in $H^{1} (R^{N}, R)$ , $v_{n} \in B$ and $\tilde{f} (s) ⩽ V_{0} / k_{0}$ , we obtain $\begin{matrix} min {1, \frac{V_{0}}{2}} (\int | \nabla v_{n} |^{2} + | v_{n} |^{2}) ⩽ o_{n} (1), \end{matrix}$ from where it follows that $\begin{matrix} v_{n} \to 0 in H^{1} (R^{N}, R), \end{matrix}$ showing that $v = 0$ , which is an absurd. This absurd shows that $(y_{n})$ has a bounded subsequence. Thus, up to a subsequence, we have that $\begin{matrix} y_{n} \to y_{0} \in R^{N} . \end{matrix}$ If $y_{0} \notin \overline{Ω}$ , we proceed as above to conclude that $v_{n} \to 0$ . Therefore, we must have $y_{0} \in \overline{Ω}$ .

In order to prove that $V (y_{0}) = V_{0}$ , we will suppose by contradiction that $V (y_{0}) > V_{0}$ . Combining (2.1) with the strong convergence of $(w_{n})$ in $H^{1} (R^{N}, R)$ , Fatou’s lemma and the invariance of $R^{N}$ by translations, we obtain $\begin{array}{rcl} c_{0} & = & E_{0} (w) < \frac{1}{2} \int (| \nabla w |^{2} + V (y_{0}) | w |^{2}) - \frac{1}{4} \int K (w) F (| w |^{2}) \\ ⩽ & \underset{n \to \infty}{lim inf} [\frac{1}{2} \int (| \nabla w_{n} |^{2} + V (ε_{n} z + {\tilde{y}}_{n}) | w_{n} |^{2}) - \frac{1}{4} \int K (w_{n}) F (| w_{n} |^{2})] \\ ⩽ & \underset{n \to \infty}{lim inf} [\frac{1}{2} \int ({| \nabla | w_{n} | |}^{2} + V (ε_{n} z + {\tilde{y}}_{n}) | w_{n} |^{2}) - \frac{1}{4} \int K (w_{n}) F (| w_{n} |^{2})] . \end{array}$ By diamagnetic inequality (2.1), $\begin{array}{rcl} c_{0} & < & \underset{n \to \infty}{lim inf} [\frac{1}{2} \int (| \nabla_{ε_{n}} w_{n} |^{2} + V (ε_{n} z + {\tilde{y}}_{n}) | w_{n} |^{2}) - \frac{1}{4} \int K (w_{n}) F (| w_{n} |^{2})] \\ ⩽ & \underset{n \to \infty}{lim inf} J_{ε_{n}} (t_{n} u_{n}) ⩽ \underset{n \to \infty}{lim inf} J_{ε_{n}} (u_{n}) = c_{0}, \end{array}$ which does not make sense. Hence $V (y_{0}) = V_{0}$ and $y_{0} \in \overline{Ω}$ . The condition $(V_{2})$ implies that $y_{0} \notin \partial Ω$ , that is, $y_{0} \in M$ , finishing the proof. □
Corollary 3.5.
Assume the same hypotheses of Lemma* 3.4. Then, for any given $γ > 0$ , there exists $R > 0$ and $n_{0} \in N$ such that $\begin{matrix} \int_{B_{R} {({\tilde{y}}_{n})}^{c}} ({| \nabla | u_{n} | |}^{2} + | u_{n} |^{2}) < γ for all n ⩾ n_{0} . \end{matrix}$
Proof.
Setting $v_{n} (x) = | u_{n} | (\cdot + {\tilde{y}}_{n})$ , for each $R > 0$ , we have that $\begin{matrix} \int_{B_{R} {({\tilde{y}}_{n})}^{c}} ({| \nabla | u_{n} | |}^{2} + | u_{n} |^{2}) = \int_{B_{R} {(0)}^{c}} (| \nabla v_{n} |^{2} + | v_{n} |^{2}) . \end{matrix}$ Now, result follows applying Lemma 3.4, because $(v_{n})$ is strongly convergent in $H^{1} (R^{N}, R)$ . □

We finalize the section presenting a relation between the topology of M and the number of solutions of the modified problem ( D ˜ ε ) a . Theorem 3.6.
For any $δ > 0$ verifying $M_{δ} \subset Ω$ , there exists ${\hat{ε}}_{δ} > 0$ such that, for any $0 < ε < {\hat{ε}}_{δ}$ , the problem ( D ˜ ε ) a has at least ${cat}_{M_{δ}} (M)$ nontrivial solutions.
Proof.
Given $δ > 0$ such that $M_{δ} \subset Ω$ , we can use (3.6), Lemma 3.2, Proposition 3.3, and argue as in [14, Section 6] to obtain ${\hat{ε}}_{δ} > 0$ such that, for any $ε \in (0, {\hat{ε}}_{δ})$ , the diagram $\begin{matrix} M \overset{Φ_{ε}}{⟶} {\tilde{N}}_{ε} \overset{β_{ε}}{⟶} M_{δ} \end{matrix}$ is well defined and $β_{ε} \circ Φ_{ε}$ is homotopically equivalent to the embedding $ι : M \to M_{δ}$ . Thus $\begin{matrix} {cat}_{{\tilde{N}}_{ε}} ({\tilde{N}}_{ε}) ⩾ {cat}_{M_{δ}} (M) . \end{matrix}$ It follows from Proposition 2.8 and standard Ljusternik–Schnirelmann theory that $J_{ε}$ possesses at least ${cat}_{{\tilde{N}}_{ε}} ({\tilde{N}}_{ε})$ critical points on $N_{ε}$ . The same argument employed in the proof of Proposition 2.8 shows that each of these critical points is also a critical point of the unconstrained functional $J_{ε}$ . Thus, we obtain ${cat}_{M_{δ}} (M)$ nontrivial solutions for ( D ˜ ε ) a . □

4. Proof of Theorem 1.1

In this section we prove our main theorem. The idea is to show that the solutions obtained in Theorem 3.6 verify the following estimate $| u_{ε} (x) | ⩽ t_{a}$ $\forall x \in Ω_{ε}^{c}$ as ε is small enough. This fact implies that these solutions are in fact solutions of the original problem ( D ε ). The key ingredient is the following result, whose proof uses an adaptation of the arguments found in [11] and [26], which are related with the Moser iteration method [37].

Lemma 4.1.

Let $ε_{n} \to 0^{+}$ and $u_{n} \in {\tilde{N}}_{ε_{n}}$ be a solution of ${({\tilde{D}}_{ε_{n}})}_{a}$ . Then $J_{ε_{n}} (u_{n}) \to c_{0}$ and $| u_{n} | \in L^{\infty} (R^{N})$ . Moreover, for any given $γ > 0$ , there exists $R > 0$ and $n_{0} \in N$ such that $\begin{matrix} (4.1) & ∥ u_{n} ∥_{L^{\infty} (B_{R} {({\tilde{y}}_{n})}^{c})} < γ for all n ⩾ n_{0}, \end{matrix}$ where ${\tilde{y}}_{n}$ is given by Lemma 3.4 .

Proof.

Since $J_{ε_{n}} (u_{n}) ⩽ c_{0} + h (ε_{n})$ with ${lim}_{n \to \infty} h (ε_{n}) = 0$ , arguing as in the proof of Eq. (3.10), we deduce that $J_{ε_{n}} (u_{n}) \to c_{0}$ . Invoking Lemma 3.4, we find a sequence $({\tilde{y}}_{n}) \subset R^{N}$ such that $ε_{n} {\tilde{y}}_{n} \to y_{0} \in M$ and $v_{n} : = | u_{n} | (\cdot + {\tilde{y}}_{n})$ has a convergent subsequence in $H^{1} (R^{N}, R)$ .

Next, we fix $R > 1$ and $η_{R} \in C^{\infty} (R^{N}, R)$ verifying $\begin{matrix} 0 ⩽ η_{R} ⩽ 1, η_{R} \equiv 0 in B_{R / 2} (0), η_{R} \equiv 1 in B_{R} {(0)}^{c} and | \nabla η_{R} | ⩽ C / R . \end{matrix}$ Moreover, for each $n \in N$ and $L > 0$ , we define $\begin{matrix} η_{n} (x) : = η_{R} (x - {\tilde{y}}_{n}), u_{L, n} (x) : = min {| u_{n} (x) |, L} and z_{L, n} : = η_{n}^{2} u_{L, n}^{2 (β - 1)} u_{n}, \end{matrix}$ with $β > 1$ to be determined later on.

By using the calculation performed in [11, Eq. (2.2)] and the diamagnetic inequality, we derive that $\begin{array}{rcl} Re (\nabla_{ε} u_{n} \overline{\nabla_{ε} z_{L, n}}) & ⩾ & η_{n}^{2} u_{L, n}^{2 (β - 1)} | \nabla_{ε} u_{n} |^{2} + 2 η_{n} | u_{n} | u_{L, n}^{2 (β - 1)} \nabla η_{n} \cdot \nabla | u_{n} | \\ ⩾ & η_{n}^{2} u_{L, n}^{2 (β - 1)} {| \nabla | u_{n} | |}^{2} + 2 η_{n} | u_{n} | u_{L, n}^{2 (β - 1)} \nabla η_{n} \cdot \nabla | u_{n} | . \end{array}$ This inequality, the definition of $z_{L, n}$ and $J_{ε_{n}}^{'} (u_{n}) z_{L, n} = 0$ guarantee that $\begin{array}{l} \int η_{n}^{2} u_{L, n}^{2 (β - 1)} {| \nabla | u_{n} | |}^{2} + 2 \int η_{n} | u_{n} | u_{L, n}^{2 (β - 1)} \nabla η_{n} \cdot \nabla | u_{n} | \\ (4.2) & ⩽ \int (K (u_{n}) g_{ε_{n}} (x, | u_{n} |^{2}) - V_{ε_{n}} (x)) η_{n}^{2} | u_{n} |^{2} u_{L, n}^{2 (β - 1)} . \end{array}$

In view of $(g_{2})$ and $(g_{3})$ , we know that there is $C_{1} > 0$ verifying $\begin{matrix} g (x, s) ⩽ \frac{V_{0}}{2} + C_{1} | s |^{(2^{*} - 2) / 2} for any (x, s) \in R^{N} \times R . \end{matrix}$ Combining (4.2) with $u_{n} \in B$ and $V_{ε} (x) ⩾ V_{0}$ , we get $\begin{array}{rcl} \int η_{n}^{2} u_{L, n}^{2 (β - 1)} {| \nabla | u_{n} | |}^{2} & ⩽ & 2 \int η_{n} | u_{n} | u_{L, n}^{2 (β - 1)} | \nabla η_{n} | | \nabla | u_{n} | | \\ + \int (\frac{V_{0}}{2} + C_{1} | u_{n} |^{2^{*} - 2} - V_{ε_{n}} (x)) η_{n}^{2} | u_{n} |^{2} u_{L, n}^{2 (β - 1)} \\ ⩽ & 2 \int η_{n} | u_{n} | u_{L, n}^{2 (β - 1)} | \nabla η_{n} | | \nabla | u_{n} | | + C_{1} \int η_{n}^{2} | u_{n} |^{2^{*}} u_{L, n}^{2 (β - 1)} . \end{array}$ Now, (4.1) follows arguing as in the proof of [4, Lemma 4.1]. □

Finally, we are ready to prove the main result of the paper.

Proof of Theorem 1.1.

Suppose that $δ > 0$ is such that $M_{δ} \subset Ω$ . We first claim that there exists ${\tilde{ε}}_{δ} > 0$ such that, for any $0 < ε < {\tilde{ε}}_{δ}$ and any solution $u \in {\tilde{N}}_{ε}$ of the problem ( D ˜ ε ) a , there holds $\begin{matrix} (4.3) & ∥ u ∥_{L^{\infty} (R^{N} ∖ Ω_{ε})} < t_{a} . \end{matrix}$ To prove the above claim, we argue by contradiction. Assume that there are two sequences $ε_{n} \to 0^{+}$ and $u_{n} \in {\tilde{N}}_{ε_{n}}$ verifying $J_{ε_{n}}^{'} (u_{n}) = 0$ and $\begin{matrix} (4.4) & ∥ u_{n} ∥_{L^{\infty} (R^{N} ∖ Ω_{ε_{n}})} ⩾ t_{a} . \end{matrix}$ As in Lemma 4.1, we have that $J_{ε_{n}} (u_{n}) \to c_{0}$ and therefore we can use Lemma 3.4 to obtain a sequence $({\tilde{y}}_{n}) \subset R^{N}$ such that $ε_{n} {\tilde{y}}_{n} \to y_{0} \in M$ .

If we take $r > 0$ such that $B_{r} (y_{0}) \subset B_{2 r} (y_{0}) \subset Ω$ we have that $\begin{matrix} B_{r / ε_{n}} (y_{0} / ε_{n}) = \frac{1}{ε_{n}} B_{r} (y_{0}) \subset Ω_{ε_{n}} . \end{matrix}$ Moreover, for any $z \in B_{r / ε_{n}} ({\tilde{y}}_{n})$ , there holds $\begin{matrix} | z - \frac{y_{0}}{ε_{n}} | ⩽ | z - {\tilde{y}}_{n} | + | {\tilde{y}}_{n} - \frac{y_{0}}{ε_{n}} | < \frac{1}{ε_{n}} (r + o_{n} (1)) < \frac{2 r}{ε_{n}} \end{matrix}$ for n large. For this values of n we have that $B_{r / ε_{n}} ({\tilde{y}}_{n}) \subset Ω_{ε_{n}}$ or, equivalently, $R^{N} ∖ Ω_{ε_{n}} \subset R^{N} ∖ B_{r / ε_{n}} ({\tilde{y}}_{n})$ . On the other hand, it follows from Lemma 4.1 with $γ = t_{a}$ that, for any $n ⩾ n_{0}$ such that $r / ε_{n} > R$ , there holds $\begin{matrix} ∥ u_{n} ∥_{L^{\infty} (R^{N} ∖ Ω_{ε_{n}})} ⩽ ∥ u_{n} ∥_{L^{\infty} (R^{N} ∖ B_{r / ε_{n}} ({\tilde{y}}_{n}))} ⩽ ∥ u_{n} ∥_{L^{\infty} (R^{N} ∖ B_{R} ({\tilde{y}}_{n}))} < t_{a}, \end{matrix}$ which contradicts (4.4) and proves the claim.

Let ${\hat{ε}}_{δ} > 0$ given by Theorem 3.6 and set $ε_{δ} : = min {{\hat{ε}}_{δ}, {\tilde{ε}}_{δ}}$ . We will show the theorem for this choice of $ε_{δ}$ . Let $0 < ε < ε_{δ}$ be fixed. By Theorem 3.6, there is ${cat}_{M_{δ}} (M)$ nontrivial solutions of the problem ( D ˜ ε ) a . If $u \in H_{ε}$ is one of these solutions, we have that $u \in {\tilde{N}}_{ε}$ . Then, by (4.3), $g_{ε} (\cdot, | u |^{2}) \equiv f (| u |^{2})$ , showing that u is a solution of the problem ( D ε ). An easy calculation shows that $\hat{u} (x) : = u (x / ε)$ is a solution of the original problem ( P ε ). Then, ( P ε ) has at least ${cat}_{M_{δ}} (M)$ nontrivial solutions.

We now consider $ε_{n} \to 0^{+}$ and take a sequence $u_{n} \in H_{ε_{n}}$ of solutions of the problem $(D_{ε_{n}})$ as above. In order to study the behavior of the maximum points of $| u_{n} |$ , we first notice that, by $(g_{2})$ , there exists $γ > 0$ such that $\begin{matrix} (4.5) & g (ε x, s^{2}) s^{2} ⩽ \frac{V_{0}}{2} s^{2} for all x \in R^{N}, | s | ⩽ γ . \end{matrix}$ By Lemma 4.1, there are $R > 0$ and $({\tilde{y}}_{n}) \subset R^{N}$ such that $\begin{matrix} (4.6) & ∥ u_{n} ∥_{L^{\infty} {(B_{R} ({\tilde{y}}_{n}))}^{c}} < γ . \end{matrix}$ Up to a subsequence, we can assume that $\begin{matrix} (4.7) & ∥ u_{n} ∥_{L^{\infty} (B_{R} ({\tilde{y}}_{n}))} ⩾ γ . \end{matrix}$ Indeed, if this is not the case, we have $∥ u_{n} ∥_{L^{\infty} (R^{N})} < γ$ , and therefore, it follows from $J_{ε_{n}}^{'} (u_{n}) = 0$ , (4.5), $u_{n} \in B$ and the diamagnetic inequality that $\begin{matrix} \int {| \nabla | u_{n} | |}^{2} + V_{0} | u_{n} |^{2} ⩽ ∥ u_{n} ∥_{ε_{n}}^{2} = \int K (u_{n}) g_{ε_{n}} (x, | u_{n} |^{2}) | u_{n} |^{2} ⩽ \frac{V_{0}}{2} \int | u_{n} |^{2} . \end{matrix}$ The above expression implies that $∥ | u_{n} | ∥_{H^{1} (R^{N}, R)} = 0$ , which does not make sense. Thus, (4.7) holds.

By (4.6) and (4.7), the maximum point $p_{n} \in R^{N}$ of $| u_{n} |$ belongs to $B_{R} ({\tilde{y}}_{n})$ . Hence $p_{n} = {\tilde{y}}_{n} + q_{n}$ for some $q_{n} \in B_{R} (0)$ . Recalling that the associated solution of $(P_{ε_{n}})$ is of the form ${\hat{u}}_{n} (x) = u_{n} (x / ε_{n})$ , we conclude that the maximum point $η_{n}$ of $| {\hat{u}}_{n} |$ is $η_{n} : = ε_{n} {\tilde{y}}_{n} + ε_{n} q_{n}$ . Since $(q_{n}) \subset B_{R} (0)$ is bounded and $ε_{n} {\tilde{y}}_{n} \to y_{0} \in M$ (according to Lemma 3.4), we see that $\begin{matrix} lim_{n \to \infty} V (η_{ε_{n}}) = V (y_{0}) = V_{0}, \end{matrix}$ concluding the proof of the theorem. □

Footnotes

Acknowledgements

C.O. Alves was partially by CNPq/Brazil 301807/2013-2; G.M. Figueiredo was partially supported by CNPq/Brazil 301292/2011-9 and 552101/2011-7; M. Yang was partially supported by NSFC (11101374), ZJNSF (LY15A010010) and CNPq/Brazil 500001/2013-8.

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