Multiple solutions for singularly perturbed nonlinear magnetic Schrödinger equations

Abstract

In this paper we consider singularly perturbed nonlinear Schrödinger equations with electromagnetic potentials and involving continuous nonlinearities with subcritical, critical or supercritical growth. By means of suitable variational techniques, truncation arguments and Lusternik–Schnirelman theory, we relate the number of nontrivial complex-valued solutions with the topology of the set where the electric potential attains its minimum value.

Keywords

Magnetic Laplacian Variational methods Lusternik–Schnirelman theory

1. Introduction

In the first part of this paper, we focus our attention on the following magnetic Schrödinger equation: $\begin{matrix} (1.1) & \{\begin{matrix} {(\frac{ε}{ı} \nabla - A (x))}^{2} u + V (x) u = f (| u |^{2}) u + γ | u |^{2^{*} - 2} u in R^{N}, \\ | u | \in H^{1} (R^{N}, R), \end{matrix} \end{matrix}$ where $ε > 0$ is a small parameter, $N ⩾ 3$ , $2^{*} = \frac{2 N}{N - 2}$ is the critical exponent, $γ \in {0, 1}$ , $A : R^{N} \to R^{N}$ is a $C^{1}$ magnetic potential, $V : R^{N} \to R$ is a continuous electric potential, and $f : R \to R$ is a continuous function with subcritical growth at infinity. The operator ${(\frac{ε}{ı} \nabla - A (x))}^{2}$ is the so called magnetic Laplacian defined as $\begin{matrix} {(\frac{ε}{ı} \nabla - A (x))}^{2} u = - ε^{2} Δ u - \frac{2 ε}{ı} A (x) \cdot \nabla u + {| A (x) |}^{2} u - \frac{ε}{ı} u div (A (x)) . \end{matrix}$ When $N = 3$ , the magnetic field B represents the curl of A, while for higher dimensions $N ⩾ 4$ , it is the 2-form given by $B_{i j} = \partial_{j} A_{k} - \partial_{k} A_{j}$ ; see [16]. Equation (1.1) arises when we seek standing waves solutions $Ψ (x, t) = u (x) e^{- \frac{ı c t}{ℏ}}$ , where ℏ is the Planck’s constant and $E \in R$ , for the following time-dependent Schrödinger equation: $\begin{matrix} (1.2) & ı ℏ \frac{\partial Ψ}{\partial t} = {(\frac{ε}{ı} \nabla - A (x))}^{2} Ψ + U (z) ψ - g (x, | Ψ |^{2}) Ψ in R^{N} \times R, \end{matrix}$ Clearly, Ψ solves (1.2) if and only if u satisfies (1.1), with $ε = ℏ$ , $V (x) = U (x) - E$ and $g (x, | u |^{2}) u = f (| u |^{2}) u + γ | u |^{2^{*} - 2} u$ . We recall that the linear Schrödinger equation plays a fundamental role in quantum mechanics and describes the dynamics of a particle in a non-relativistic setting. The nonlinear Schrödinger equation appears in different physical theories, e.g., the description of Bose–Einstein condensates and nonlinear optics; see [16,41] for more physical background. An important class of solutions of (1.1) are the so called semiclassical states which concentrate and develop a spike shape around one, or more, particular points in $R^{N}$ , while vanishing elsewhere as $ε \to 0$ . The interest for semiclassical states is due to the well-known fact that the transition from quantum mechanics to classical mechanics can be formally performed by sending $ε \to 0$ .

In recent years, much attention has been paid to nonlinear Schrödinger equations without the magnetic vector potential (i.e. $A (x) \equiv 0$ ) for studying the existence, multiplicity and qualitative properties of standing wave solutions; see for instance [2–4,17,22,23,26,28,30,39,42,43]. If the magnetic vector potential $A (x) ≢ 0$ , it seems that the first work was due to Esteban and Lions [25] who studied the existence of a ground state solution in dimensions $N = 2$ or $N = 3$ , by using the concentration-compactness principle [35] and minimization arguments. Later, Kurata [32] proved, via variational methods, that a subcritical magnetic Schrödinger equation has a least energy solution for any $ε > 0$ , assuming a technical condition linking V and A. Chabrowski and Szulkin [19] applied minimax arguments to deduce the existence of nontrivial solutions for a critical magnetic Schrödinger equation when the potential V changes sign. Cingolani [21] used Lusternik–Schnirelman theory to obtain multiple solutions for a subcritical magnetic Schrödinger equation assuming the following global condition on the potential V introduced by Rabinowitz [39]: $\begin{matrix} inf_{x \in R^{N}} V (x) < \underset{| x | \to \infty}{lim inf} V (x) . \end{matrix}$ Alves et al. [6] combined penalization technique and Lusternik–Schnirelman theory to relate the number of solutions of (1.1) with the topology of the set where the potential attains its minimum value, by considering (1.1) with $γ = 0$ , $f \in C^{1}$ has a subcritical growth, and assuming that the potential $V : R^{N} \to R$ is a continuous function satisfying the following assumptions motivated by del Pino and Felmer [23]:

there exists $V_{0} > 0$ such that $V_{0} = {inf}_{x \in R^{N}} V (x)$ ;

there exists a bounded open set $Λ \subset R^{N}$ such that $\begin{matrix} (1.3) & V_{0} < min_{\partial Λ} V and M = {x \in Λ : V (x) = V_{0}} \neq \emptyset . \end{matrix}$

In [8], the author dealt with the existence and concentration phenomenon as

ε \to 0

of nontrivial solutions to (1.1) when

γ = 1

. We also mention [5,15,24,45] for other interesting results on magnetic Schrödinger equations, and [9–14,36] for some recent existence and multiplicity results for fractional magnetic Schrödinger type equations.

Particularly motivated by [6,8,9,23], in this paper we investigate the multiplicity and concentration of nontrivial solutions to (1.1) when $ε \to 0$ , under assumptions $(V_{1})$ – $(V_{2})$ and by considering continuous nonlinearities with subcritical, critical or supercritical growth. More precisely, we first consider the case when the nonlinearity $f : R \to R$ is a continuous function fulfilling the following conditions:

$f (t) = 0$ for $t ⩽ 0$ ;

if $γ = 0$ , there exists $q \in (2, 2^{*})$ such that $\begin{matrix} lim_{t \to \infty} \frac{f (t)}{t^{\frac{q - 2}{2}}} = 0, \end{matrix}$ if $γ = 1$ , there exist $λ > 0$ and $q_{1}, σ \in (2, 2^{*})$ such that $\begin{array}{l} f (t) ⩾ λ t^{\frac{q_{1} - 2}{2}} for all t > 0, lim_{t \to \infty} \frac{f (t)}{t^{\frac{σ - 2}{2}}} = 0, \end{array}$ where λ is such that

$λ > 0$ if either $N ⩾ 4$ , or $N = 3$ and $4 < q < 6$ ,

λ is sufficiently large if $N = 3$ and $2 < q ⩽ 4$ ,

there exists $θ \in (2, 2^{*})$ such that $0 < \frac{θ}{2} F (t) ⩽ t f (t)$ for any $t > 0$ , where $F (t) = \int_{0}^{t} f (τ) d τ$ ;

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

Then, our first main result can be stated as follows:

Theorem 1.1.
Assume that $(V_{1})$ – $(V_{2})$ and $(f_{1})$ – $(f_{4})$ hold. Then, for any $δ > 0$ such that $\begin{matrix} M_{δ} = {x \in R^{N} : dist (x, M) ⩽ δ} \subset Λ, \end{matrix}$ there exists $ε_{δ} > 0$ such that, for any $ε \in (0, ε_{δ})$ , problem ( 1.1 ) has at least ${cat}_{M_{δ}} (M)$ nontrivial solutions. Moreover, if $u_{ε}$ denotes one of these solutions and $x_{ε}$ is a global maximum point of $| u_{ε} |$ , then we have $\begin{matrix} lim_{ε \to 0} V (x_{ε}) = V_{0}, \end{matrix}$ and there exist $C_{1}, C_{2} > 0$ such that $\begin{matrix} | u_{ε} (x) | ⩽ C_{1} e^{- \frac{C_{2}}{ε} | x - x_{ε} |} for all x \in R^{N} . \end{matrix}$

The proof of Theorem 1.1 will be obtained by combining variational techniques, a penalization argument and Lusternik–Schnirelman theory. First of all, due to the lack of information on the behavior of V at infinity, we adapt the penalization argument in [23], which consists in making a suitable modification on f, solving a modified problem and then check that, for $ε > 0$ small enough, the solutions of the modified problem are indeed solutions of the original one. To obtain multiple solutions of the modified problem, we use a technique introduced by Benci and Cerami in [18]. The main ingredient is to make precisely comparisons between the category of some sublevel sets of the modified functional and the category of the set M. However, the nonlinearity f is only continuous, so we cannot use standard arguments on $C^{1}$ -Nehari manifold as in [3,4,6,44]. To overcome the non differentiability of the Nehari manifold, we take advantage of some variants of critical point theorems from Szulkin and Weth [42] (see also [27]). We stress that our arguments are totally distinct from [6,8,27], and our theorem improves and complements the results in [6,8] because we are considering the multiplicity to (1.1) with continuous nonlinearities with subcritical or critical growth. Indeed, we believe that the ideas contained here can be also used in other situations to deal with local conditions on the potential V and the non-differentiability of the nonlinearity f. Finally, we prove the exponential decay of modulus of solutions of (1.1) by using Kato’s inequality [31] and a comparison argument.

In the second part of this work, we consider the following magnetic supercritical equation: $\begin{matrix} (1.4) & {(\frac{ε}{ı} \nabla - A (x))}^{2} u + V (x) u = | u |^{q - 2} u + λ | u |^{r - 2} u in R^{N}, \end{matrix}$ where $ε > 0$ , $λ > 0$ , and $2 < q < 2^{} < r$ . In this case we are able to prove the next result:
Theorem 1.2.
Suppose that V satisfies* $(V_{1})$ – $(V_{2})$ . Then there exists $λ_{0} > 0$ with the following property: for any $λ \in (0, λ_{0})$ and $δ > 0$ given, there exists $ε_{λ, δ} > 0$ such that, for any $ε \in (0, ε_{λ, δ})$ , problem ( 1.4 ) has at least ${cat}_{M_{δ}} (M)$ nontrivial solutions.

Since $r > 2^{*}$ , we cannot directly use variational techniques because the functional associated with (1.4) is not well defined on the magnetic Sobolev space $H_{ε}$ (see Section 2 for its definition). In order to overcome this difficulty, inspired by [20,26,38], we consider a truncated problem with subcritical growth for which we can apply Theorem 1.1 and we obtain a multiplicity result for the truncated problem. After proving a priori bounds (independent of λ) for these solutions, we use a suitable Moser iteration scheme [37] to verify that, if λ is sufficiently small, the solutions of the truncated problem also satisfy the original problem (1.4). To the best of our knowledge, in the literature there are no multiplicity results for supercritical magnetic problems via Lusternik–Schnirelman theory.

The paper is organized as follows. In Section 2, we collect some preliminary results which will be used along the paper. In Section 3 we study the modified functional. In Section 4, we consider autonomous scalar problem related to (1.1) and we provide a multiplicity result for the modified problem. In Section 5, we present the proof of Theorem 1.1. In Section 6, we deal with the supercritical case.
2. Preliminaries

Let us denote by $D^{1, 2} (R^{N}, R)$ the closure of $C_{c}^{\infty} (R^{N}, R)$ with respect to $\begin{matrix} ‖ \nabla u ‖_{L^{2} (R^{N})} = {(\int_{R^{N}} | \nabla u |^{2} d x)}^{\frac{1}{2}} . \end{matrix}$ When $N ⩾ 3$ , we also know that $D^{1, 2} (R^{N}, R) = {u \in L^{2^{*}} (R^{N}, R) : ‖ \nabla u ‖_{L^{2} (R^{N})} < \infty}$ . We denote by $H^{1} (R^{N}, R)$ the Sobolev space $\begin{matrix} H^{1} (R^{N}, R) = {u : R^{N} \to R : u, \nabla u \in L^{2} (R^{N}, R)} . \end{matrix}$ We recall that there exists a sharp constant $S_{*} > 0$ such that for any $u \in D^{1, 2} (R^{N}, R)$ $\begin{matrix} ‖ u ‖_{L^{2^{*}} (R^{N})}^{2} ⩽ S_{*}^{- 1} ‖ \nabla u ‖_{L^{2} (R^{N})}^{2} . \end{matrix}$ Moreover, the embedding $H^{1} (R^{N}, R) \subset L^{q} (R^{N}, R)$ is continuous for all $q \in [2, 2^{*})$ and locally compact for all $q \in [1, 2^{*})$ ; see [1]. We also recall the following vanishing lemma.

Lemma 2.1 ([34]).

Let $q \in [2, 2^{*})$ . If $(u_{n})$ is a bounded sequence in $H^{1} (R^{N}, R)$ and if $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{N}} \int_{B_{R} (y)} | u_{n} |^{q} d x = 0 \end{matrix}$ for some $R > 0$ , then $u_{n} \to 0$ in $L^{r} (R^{N}, R)$ for all $r \in (2, 2^{*})$ .

Let $L^{2} (R^{N}, C)$ be the space of complex-valued functions such that $‖ u ‖_{L^{2} (R^{N})}^{2} = \int_{R^{N}} | u |^{2} d x < \infty$ endowed with the inner product ${⟨ u, v ⟩}_{L^{2}} = ℜ \int_{R^{N}} u \bar{v} d x$ , where $ℜ (z)$ denotes the real part of $z \in C$ and $\bar{z}$ is its conjugate. In order to study (1.1), for $ε > 0$ , we introduce the Hilbert space $H_{ε}$ obtained by the closure of $C_{c}^{\infty} (R^{N}, C)$ under the inner product $\begin{matrix} {⟨ u, v ⟩}_{ε} = ℜ (\int_{R^{N}} \nabla_{ε} u \overline{\nabla_{ε} v} + V (ε x) u \bar{v} d x), \end{matrix}$ where $\nabla_{ε} u = (D_{ε}^{1} u, \dots, D_{ε}^{N} u)$ and $D_{ε}^{j} u = ı^{- 1} \partial_{j} u - A_{j} (ε x) u$ for $j = 1, \dots, N$ . The norm induced by this inner product is given by $\begin{matrix} ‖ u ‖_{ε} = {(\int_{R^{N}} | \nabla_{ε} u |^{2} + V (ε x) | u |^{2} d x)}^{\frac{1}{2}} . \end{matrix}$ As proved in [25], for any $u \in H_{ε}$ it holds the following diamagnetic inequality: $\begin{matrix} (2.1) & | \nabla | u | (x) | ⩽ | \nabla_{ε} u (x) | for a.e. x \in R^{N} . \end{matrix}$ Consequently, if $u \in H_{ε}$ then $| u | \in H^{1} (R^{N}, R)$ . Moreover, the embedding $H_{ε} \subset L^{q} (R^{N}, C)$ is continuous for all $q \in [2, 2^{*}]$ and $H_{ε} \subset L^{q} (K, C)$ is compact for all $K \subset R^{N}$ compact and any $q \in [1, 2^{*})$ . We also recall the following distributional Kato’s inequality [31] (see also [40]): $\begin{matrix} (2.2) & Δ | u | ⩾ - ℜ (sign (u) D_{ε}^{2} u) for all u \in H_{ε}, \end{matrix}$ where $D_{ε}^{2} = \sum_{j = 1}^{N} {(D_{ε}^{j})}^{2}$ and $\begin{matrix} sign (u) = \{\begin{matrix} \frac{\bar{u}}{| u |} & if u \neq 0, \\ 0 & if u = 0 . \end{matrix} \end{matrix}$

3. Variational setting and the modified functional

Using the change of variable $u (x) \mapsto u (ε x)$ , we see that (1.1) is equivalent to the following problem $\begin{matrix} (3.1) & \{\begin{matrix} {(\frac{1}{ı} \nabla - A_{ε} (x))}^{2} u + V_{ε} (x) u = f (| u |^{2}) u + γ | u |^{2^{*} - 2} u in R^{N}, \\ | u | \in H^{1} (R^{N}, R), \end{matrix} \end{matrix}$ where we set $A_{ε} (x) = A (ε x)$ and $V_{ε} (x) = V (ε x)$ . Without loss of generality, we assume that $0 \in M$ . Fix $k > 1$ and $a > 0$ such that $f (a) + γ a^{\frac{2^{*} - 2}{2}} = \frac{V_{0}}{k}$ , and we consider the function $\begin{matrix} \tilde{f} (t) = \{\begin{matrix} f (t) + γ {(t^{+})}^{\frac{2^{*} - 2}{2}} & if t ⩽ a, \\ \frac{V_{0}}{k} & if t > a . \end{matrix} \end{matrix}$ We introduce the penalized nonlinearity $g : R^{N} \times R \to R$ by setting $\begin{matrix} g (x, t) = χ_{Λ} (x) (f (t) + γ {(t^{+})}^{\frac{2^{*} - 2}{2}}) + (1 - χ_{Λ} (x)) \tilde{f} (t), \end{matrix}$ where $χ_{Λ}$ is the characteristic function of Λ, and we set $G (x, t) = \int_{0}^{t} g (x, τ) d τ$ .

By assumptions $(f_{1})$ – $(f_{4})$ , it is easy to check that g is a Carathéodory function satisfying the following properties:

${lim}_{t \to 0} g (x, t) = 0$ uniformly in $x \in R^{N}$ ;

$g (x, t) ⩽ f (t)$ for any $x \in R^{N}$ and $t > 0$ ;

(i) $0 < \frac{θ}{2} G (x, t) ⩽ g (x, t) t$ for any $x \in Λ$ and $t > 0$ ,

(ii) $0 ⩽ G (x, t) ⩽ g (x, t) t ⩽ \frac{V (x)}{k} t$ and $0 ⩽ g (x, t) ⩽ \frac{V (x)}{k}$ for any $x \in Λ^{c}$ and $t > 0$ ;

for any $x \in Λ$ , the function $t \mapsto g (x, t)$ is increasing in $(0, \infty)$ , and for any $x \in Λ^{c}$ , the function $t \mapsto g (x, t)$ is increasing in $(0, a)$ .

Set

g_{ε} (x, t) = g (ε x, t)

and we consider the following modified problem

\begin{matrix} (3.2) & \{\begin{matrix} {(\frac{1}{ı} \nabla - A_{ε} (x))}^{2} u + V_{ε} (x) u = g_{ε} (x, | u |^{2}) u in R^{N}, \\ | u | \in H^{1} (R^{N}, R) . \end{matrix} \end{matrix}

Let us note that if u is a solution of (3.2) such that

\begin{matrix} (3.3) & | u (x) | ⩽ \sqrt{a} for all x \in Λ_{ε}^{c}, \end{matrix}

where

Λ_{ε} = {x \in R^{N} : ε x \in Λ}

, then u is also a solution of (3.1). In order to study weak solutions of (3.2), we look for critical points of the functional

J_{ε} : H_{ε} \to R

defined as

\begin{array}{l} J_{ε} (u) = \frac{1}{2} ‖ u ‖_{ε}^{2} - \frac{1}{2} \int_{R^{N}} G_{ε} (x, | u |^{2}) d x . \end{array}

It is easy to check that

J_{ε} \in C^{1} (H_{ε}, R)

and that its differential is given by

\begin{array}{l} ⟨ J_{ε}^{'} (u), v ⟩ = {⟨ u, v ⟩}_{ε} - ℜ \int_{R^{N}} g_{ε} (x, | u |^{2}) u \bar{v} d x for all u, v \in H_{ε} . \end{array}

Therefore, weak solutions to (3.2) can be found as critical points of

J_{ε}

. We start by showing that

J_{ε}

possesses a mountain pass geometry [7].

Lemma 3.1.
The functional $J_{ε}$ satisfies the following conditions:
$J_{ε} (0) = 0$ ;

there exist $α, ρ > 0$ such that $J_{ε} (u) ⩾ α$ for any $u \in H_{ε}$ such that $‖ u ‖_{ε} = ρ$ ;

there exists $e \in H_{ε}$ such that $‖ e ‖_{ε} > ρ$ and $J_{ε} (e) < 0$ .

Proof.
Clearly, $J_{ε} (0) = 0$ . By $(g_{1})$ and $(g_{2})$ , for all $δ > 0$ there exists $C_{δ} > 0$ such that $\begin{matrix} | G_{ε} (x, t^{2}) | ⩽ δ | t |^{2} + C_{δ} | t |^{2^{}} for every (x, t) \in R^{N} \times R . \end{matrix}$ This fact combined with Sobolev embeddings implies that for all $u \in H_{ε}$ $\begin{matrix} J_{ε} (u) ⩾ (\frac{1}{2} - δ C) ‖ u ‖_{ε}^{2} - C_{δ} ‖ u ‖_{ε}^{2^{}} . \end{matrix}$ Choosing $δ > 0$ sufficiently small we obtain that (ii) holds true. Now, fix $u \in H_{ε}$ such that $| supp (| u |) \cap Λ_{ε} | > 0$ . By using $(f_{3})$ , we get $\begin{array}{l} J_{ε} (T u) & ⩽ \frac{T^{2}}{2} ‖ u ‖_{ε}^{2} - \frac{1}{2} \int_{Λ_{ε}} F (T^{2} | u |^{2}) d x \\ ⩽ \frac{T^{2}}{2} ‖ u ‖_{ε}^{2} - C_{1} T^{θ} \int_{Λ_{ε}} | u |^{θ} d x + C_{2} \end{array}$ which in view of $θ > 2$ yields $J_{ε} (T u) \to - \infty$ as $T \to \infty$ . Therefore, (iii) is satisfied. □

By Lemma 3.1 it follows that we can define the minimax level $\begin{array}{l} c_{ε}^{'} = inf_{γ \in Γ_{ε}} max_{t \in [0, 1]} J_{ε} (γ (t)) where Γ_{ε} = {γ \in C ([0, 1], H_{ε}) : γ (0) = 0 and J_{ε} (γ (1)) < 0} . \end{array}$ Using a version of the mountain pass theorem without the Palais–Smale condition (see [44]), we can find a Palais–Smale sequence $(u_{n})$ at the level $c_{ε}^{'}$ .

Now, we prove that $J_{ε}$ fulfills the following compactness property:
Lemma 3.2.
The functional $J_{ε}$ satisfies the ${(P S)}_{c}$ condition at any level $c \in R$ if $γ = 0$ , and at any level $0 < c < \frac{1}{N} S_{}^{\frac{N}{2}}$ when* $γ = 1$ .
Proof.
The proof of this lemma can be obtained arguing as in Lemma 2.1 in [6] (when $γ = 0$ ) and Lemma 3.3 in [8] (when $γ = 1$ ). For completeness, we give the details. Let $(u_{n}) \subset H_{ε}$ be a Palais–Smale sequence for $J_{ε}$ at the level c, that is $\begin{matrix} J_{ε} (u_{n}) \to c and J_{ε}^{'} (u_{n}) \to 0 in H_{ε}^{} . \end{matrix}$ Step 1 The sequence $(u_{n})$ is bounded in $H_{ε}$ . Indeed, by $(g_{3})$ , we get $\begin{array}{l} C (1 + ‖ u_{n} ‖_{ε}) ⩾ & J_{ε} (u_{n}) - \frac{1}{θ} ⟨ J_{ε}^{'} (u_{n}), u_{n} ⟩ \\ = & (\frac{θ - 2}{2 θ}) ‖ u_{n} ‖_{ε}^{2} + \frac{1}{θ} \int_{Λ_{ε}^{c}} [g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} - \frac{θ}{2} G_{ε} (x, | u_{n} |^{2})] d x \\ + \frac{1}{θ} \int_{Λ_{ε}} [f (| u_{n} |^{2}) | u_{n} |^{2} - \frac{θ}{2} F (| u_{n} |^{2})] d x + γ (\frac{1}{θ} - \frac{1}{2^{}}) \int_{Λ_{ε}} | u_{n} |^{2^{}} d x \\ ⩾ & (\frac{θ - 2}{2 θ}) ‖ u_{n} ‖_{ε}^{2} + \frac{1}{θ} \int_{Λ_{ε}^{c}} [g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} - \frac{θ}{2} G_{ε} (x, | u_{n} |^{2})] d x \\ ⩾ & (\frac{θ - 2}{2 θ}) ‖ u_{n} ‖_{ε}^{2} - (\frac{θ - 2}{2 θ}) \frac{1}{k} \int_{Λ_{ε}^{c}} V_{ε} (x) | u_{n} |^{2} d x \\ ⩾ & (\frac{θ - 2}{2 θ}) (1 - \frac{1}{k}) ‖ u_{n} ‖_{ε}^{2}, \end{array}$ and using the facts that $k > 1$ and $θ > 2$ , we deduce that $(u_{n})$ is bounded in $H_{ε}$ . Consequently, we may assume that $u_{n} ⇀ u$ in $H_{ε}$ .

Step 2 For any $ξ > 0$ there exists $R = R_{ξ} > 0$ such that $Λ_{ε} \subset B_{\frac{R}{2}}$ and $\begin{matrix} (3.4) & \underset{n \to \infty}{lim sup} \int_{B_{R}^{c}} (| \nabla_{ε} u_{n} |^{2} + V_{ε} (x) {| u_{n} (x) |}^{2}) d x < ξ . \end{matrix}$ Let us consider the function $η_{R} \in C^{\infty} (R^{N}, R)$ defined as $\begin{matrix} η_{R} (x) = \{\begin{matrix} 0 & if x \in B_{\frac{R}{2}}, \\ 1 & if x \notin B_{R}, \end{matrix} \end{matrix}$ $0 ⩽ η_{R} ⩽ 1$ and $‖ \nabla η_{R} ‖_{L^{\infty} (R^{N})} ⩽ C / R$ for some $C > 0$ independent of R. Take $R > 0$ sufficiently large such that $Λ_{ε} \subset B_{\frac{R}{2}}$ . Since $⟨ J_{ε}^{'} (u_{n}), η_{R} u_{n} ⟩ = o_{n} (1)$ and $\begin{matrix} \overline{\nabla_{ε} (u_{n} η_{R})} = ı {\overline{u}}_{n} \nabla η_{R} + η_{R} \overline{\nabla_{ε} u_{n}}, \end{matrix}$ we can use $(g_{3})$ to see that $\begin{array}{l} \int_{R^{N}} (| \nabla_{ε} u_{n} |^{2} + V_{ε} (x) | u_{n} |^{2}) η_{R} d x \\ ⩽ \frac{1}{k} \int_{R^{N}} V_{ε} (x) | u_{n} |^{2} η_{R} d x + ℜ (\int_{R^{N}} - ı {\bar{u}}_{n} \nabla_{ε} u_{n} \nabla η_{R} d x) + o_{n} (1) . \end{array}$ By the Hölder inequality and the boundedness of $(u_{n})$ in $H_{ε}$ , we deduce that $\begin{array}{l} \int_{R^{N}} | \nabla_{ε} u_{n} |^{2} η_{R} d x + (1 - \frac{1}{k}) \int_{R^{N}} V_{ε} (x) | u_{n} |^{2} η_{R} d x ⩽ \frac{C}{R} + o_{n} (1), \end{array}$ from which $\begin{array}{l} lim_{R \to \infty} \underset{n \to \infty}{lim sup} \int_{B_{R}^{c}} (| \nabla_{ε} u_{n} |^{2} + V_{ε} (x) | u_{n} |^{2}) d x = 0, \end{array}$ that is (3.4) holds true.

Step 3 $u_{n} \to u$ in $H_{ε}$ .

We know that $u_{n} ⇀ u$ in $H_{ε}$ . Since $H_{ε}$ is compactly embedded in $L_{loc}^{r} (R^{N}, C)$ for all $r \in [1, 2^{})$ , $C_{c}^{\infty} (R^{N}, C)$ is dense in $H_{ε}$ , and using the growth assumptions on g, it is easy to prove that $⟨ J_{ε}^{'} (u), v ⟩ = 0$ for all $v \in H_{ε}$ . In particular, $\begin{matrix} ‖ u ‖_{ε}^{2} = \int_{R^{N}} g_{ε} (x, | u |^{2}) | u |^{2} d x . \end{matrix}$ Recalling that $⟨ J_{ε}^{'} (u_{n}), u_{n} ⟩ = o_{n} (1)$ , we can infer that $\begin{matrix} ‖ u_{n} ‖_{ε}^{2} = \int_{R^{N}} g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} d x + o_{n} (1) . \end{matrix}$ In view of the above relations and recalling that $H_{ε}$ is a Hilbert space, it is enough to show that $\begin{array}{l} (3.5) & lim_{n \to \infty} \int_{R^{N}} g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} d x = \int_{R^{N}} g_{ε} (x, | u |^{2}) | u |^{2} d x \end{array}$ to deduce that $u_{n} \to u$ in $H_{ε}$ . We start by considering the case $γ = 0$ . Using (3.4) and $(V_{1})$ , we see that for any $ξ > 0$ there exists $R = R_{ξ} > 0$ such that $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{B_{R}^{c}} | u_{n} |^{2} d x ⩽ C ξ . \end{matrix}$ Hence, recalling the compactness of $H_{ε}$ in $L^{2} (B_{R}, C)$ , we find $\begin{array}{l} \underset{n \to \infty}{lim sup} ‖ u_{n} - u ‖_{L^{2} (R^{N})}^{2} & = \underset{n \to \infty}{lim sup} [‖ u_{n} - u ‖_{L^{2} (B_{R})}^{2} + ‖ u_{n} - u ‖_{L^{2} (B_{R}^{c})}^{2}] \\ = lim_{n \to \infty} ‖ u_{n} - u ‖_{L^{2} (B_{R})}^{2} + \underset{n \to \infty}{lim sup} ‖ u_{n} - u ‖_{L^{2} (B_{R}^{c})}^{2} \\ = \underset{n \to \infty}{lim sup} ‖ u_{n} - u ‖_{L^{2} (B_{R}^{c})}^{2} \\ ⩽ C ξ, \end{array}$ and letting $ξ \to 0$ we deduce that $u_{n} \to u$ in $L^{2} (R^{N}, C)$ . By interpolation, $u_{n} \to u$ in $L^{p} (R^{N}, C)$ for all $p \in [2, 2^{})$ . By using $(g_{1})$ and $(g_{2})$ , we deduce that (3.5) holds true. Now we consider the case $γ = 1$ . Since $\begin{matrix} g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} ⩽ f (| u_{n} |^{2}) | u_{n} |^{2} + a^{2^{}} + \frac{V_{0}}{k} | u_{n} |^{2} in R^{N} ∖ Λ_{ε}, \end{matrix}$ and the set $B_{R} \cap Λ_{ε}^{c}$ is bounded, we can use $(f_{1})$ , $(f_{2})$ and the dominated convergence theorem to see that $\begin{matrix} (3.6) & \int_{B_{R} \cap Λ_{ε}^{c}} g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} = \int_{B_{R} \cap Λ_{ε}^{c}} g_{ε} (x, | u |^{2}) | u |^{2} d x + o_{n} (1) . \end{matrix}$ In order to show the validity of (3.5), we only need to prove that $\begin{array}{l} (3.7) & \int_{Λ_{ε}} | u_{n} |^{2^{}} d x = \int_{Λ_{ε}} | u |^{2^{}} d x + o_{n} (1) . \end{array}$ Indeed, once proved (3.7), we deduce that $\begin{matrix} \int_{B_{R} \cap Λ_{ε}} g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} = \int_{B_{R} \cap Λ_{ε}} g_{ε} (x, | u |^{2}) | u |^{2} d x + o_{n} (1) \end{matrix}$ which combined with (3.6) implies that (3.5) holds.

Using the boundedness of $(u_{n})$ in $H_{ε}$ and (2.1), we may assume that $\begin{matrix} (3.8) & {| \nabla | u_{n} | |}^{2} ⇀ μ and | u_{n} |^{2^{}} ⇀ ν \end{matrix}$ in the sense of measures. Using the concentration compactness principle [35], we obtain an at most countable index set I, sequences $(x_{i}) \subset R^{N}$ , $(μ_{i}), (ν_{i}) \subset (0, \infty)$ such that $\begin{matrix} (3.9) & μ ⩾ {| \nabla | u | |}^{2} + \sum_{i \in I} μ_{i} δ_{x_{i}}, ν = | u |^{2^{}} + \sum_{i \in I} ν_{i} δ_{x_{i}} and S_{} ν_{i}^{\frac{2}{2^{}}} ⩽ μ_{i} \forall i \in I . \end{matrix}$ Thus, it is enough to prove that $(x_{i}) \cap Λ_{ε} = \emptyset$ . Suppose by contradiction that there exists $i \in I$ such that $x_{i} \in Λ_{ε}$ . For any $ρ > 0$ , we define $ψ_{ρ} (x) = ψ ((x - x_{i}) / ρ)$ , where $ψ \in C_{c}^{\infty} (R^{N}, [0, 1])$ , $ψ = 1$ in $B_{1}$ , $ψ = 0$ in $B_{2}^{c}$ and $‖ \nabla ψ ‖_{L^{\infty} (R^{N})} ⩽ 2$ . We may assume that ρ is chosen in such a way that $supp (ψ_{ρ}) \subset Λ_{ε}$ . Since $(ψ_{ρ} u_{n})$ is bounded in $H_{ε}$ and $⟨ J_{ε}^{'} (u_{n}), ψ_{ρ} u_{n} ⟩ = o_{n} (1)$ , we can use $(V_{1})$ and (2.1) to see that $\begin{array}{l} \int_{R^{N}} {| \nabla | u_{n} | |}^{2} ψ_{ρ} d x ⩽ & \int_{R^{N}} | \nabla_{ε} u_{n} |^{2} ψ_{ρ} d x \\ ⩽ & - ℜ (\int_{R^{N}} ı {\bar{u}}_{n} \nabla_{ε} u_{n} \nabla ψ_{ρ} d x) + \int_{R^{N}} f (| u_{n} |^{2}) | u_{n} |^{2} ψ_{ρ} d x \\ (3.10) & + \int_{R^{N}} ψ_{ρ} | u_{n} |^{2^{}} d x + o_{n} (1) . \end{array}$ Since f has subcritical growth and $ψ_{ρ}$ has compact support, we get $\begin{array}{l} (3.11) & lim_{ρ \to 0} lim_{n \to \infty} \int_{R^{N}} f (| u_{n} |^{2}) | u_{n} |^{2} ψ_{ρ} d x = lim_{ρ \to 0} \int_{R^{N}} f (| u |^{2}) | u |^{2} ψ_{ρ} d x = 0 . \end{array}$ Applying the Hölder inequality, and using the boundedness of $(u_{n})$ in $H_{ε}$ , the strong convergence of $(| u_{n} |)$ in $L_{loc}^{2} (R^{N}, R)$ , $| u | \in L^{2^{}} (R^{N}, R)$ , $| \nabla ψ_{ρ} | ⩽ C ρ^{- 1}$ and $| B (x_{i}, 2 ρ) | \sim ρ^{N}$ we have $\begin{array}{l} \underset{n \to \infty}{lim sup} | \int_{R^{N}} ı {\bar{u}}_{n} \nabla_{ε} u_{n} \nabla ψ_{ρ} d x | & ⩽ \underset{n \to \infty}{lim sup} {(\int_{B_{2 ρ} (x_{i})} | u_{n} |^{2} | \nabla ψ_{ρ} |^{2} d x)}^{\frac{1}{2}} ‖ \nabla_{ε} u_{n} ‖_{L^{2} (R^{N})} \\ ⩽ C {(\int_{B_{2 ρ} (x_{i})} | u |^{2} | \nabla ψ_{ρ} |^{2} d x)}^{\frac{1}{2}} \\ ⩽ C {(\int_{B_{2 ρ} (x_{i})} | u |^{2^{}} d x)}^{\frac{1}{2^{}}} \to 0 as ρ \to 0, \end{array}$ which gives $\begin{array}{l} (3.12) & lim_{ρ \to 0} \underset{n \to \infty}{lim sup} | \int_{R^{N}} ı {\bar{u}}_{n} \nabla_{ε} u_{n} \nabla ψ_{ρ} d x | = 0 . \end{array}$ Putting together (3.8), (3.10), (3.11) and (3.12), we can conclude that $ν_{i} ⩾ μ_{i}$ . This combined with the last statement in (3.9) yields that $\begin{array}{l} (3.13) & ν_{i} ⩾ S_{}^{\frac{N}{2}} . \end{array}$ Now, in the light $(g_{3})$ and $(f_{3})$ , we obtain $\begin{array}{l} c = & J_{ε} (u_{n}) - \frac{1}{2} ⟨ J_{ε}^{'} (u_{n}), u_{n} ⟩ + o_{n} (1) \\ = & \frac{1}{2} \int_{R^{N} ∖ Λ_{ε}} [g_{ε} (x, | u_{n} |^{2}) | u_{n} |^{2} - G_{ε} (x, | u_{n} |^{2})] d x \\ + \frac{1}{2} \int_{Λ_{ε}} [f (| u_{n} |^{2}) | u_{n} |^{2} - F (| u_{n} |^{2})] d x + \frac{1}{N} \int_{Λ_{ε}} | u_{n} |^{2^{}} d x + o_{n} (1) \\ ⩾ & \frac{1}{N} \int_{Λ_{ε}} | u_{n} |^{2^{}} d x + o_{n} (1) \\ ⩾ & \frac{1}{N} \int_{Λ_{ε}} | u_{n} |^{2^{}} ψ_{ρ} d x + o_{n} (1) . \end{array}$ Taking the limit and using (3.9) and (3.13) we get $\begin{matrix} c ⩾ \frac{1}{N} \sum_{{i \in I : x_{i} \in Λ_{ε}}} ψ_{ρ} (x_{i}) ν_{i} = \frac{1}{N} \sum_{{i \in I : x_{i} \in Λ_{ε}}} ν_{i} ⩾ \frac{1}{N} S_{}^{\frac{N}{2}} \end{matrix}$ which is in contrast with $c < \frac{1}{N} S_{}^{\frac{N}{2}}$ . This ends the proof of (3.7). □
Remark 3.1.
By Lemma 3.2 in [8] we know that $0 < c_{ε}^{'} < \frac{1}{N} S_{}^{\frac{N}{2}}$ .

Since we are looking for multiple critical points of the functional $J_{ε}$ , we shall consider it constrained to an appropriated subset of $H_{ε}$ . More precisely, we introduce the Nehari manifold associated with $J_{ε}$ , namely $\begin{matrix} N_{ε} = {u \in H_{ε} ∖ {0} : ⟨ J_{ε}^{'} (u), u ⟩ = 0} . \end{matrix}$ Let us define $\begin{matrix} H_{ε}^{+} = {u \in H_{ε} : | supp (| u |) \cap Λ_{ε} | > 0} \subset H_{ε} . \end{matrix}$ Let $S_{ε}$ be the unit sphere of $H_{ε}$ and we denote by $S_{ε}^{+} = S_{ε} \cap H_{ε}^{+}$ . We observe that $H_{ε}^{+}$ is open in $H_{ε}$ . By the definition of $S_{ε}^{+}$ and the fact that $H_{ε}^{+}$ is open in $H_{ε}$ , it follows that $S_{ε}^{+}$ is a incomplete $C^{1, 1}$ -manifold of codimension 1, modeled on $H_{ε}$ and contained in the open $H_{ε}^{+}$ . Then, $H_{ε} = T_{u} S_{ε}^{+} \oplus R u$ for each $u \in S_{ε}^{+}$ , where $\begin{matrix} T_{u} S_{ε}^{+} = {v \in H_{ε} : {⟨ u, v ⟩}_{ε} = 0} . \end{matrix}$ From the growth conditions of g, we see that for any fixed $u \in N_{ε}$ and $δ > 0$ small enough $\begin{array}{l} 0 & = ⟨ J_{ε}^{'} (u), u ⟩ = ‖ u ‖_{ε}^{2} - \int_{R^{N}} g_{ε} (x, | u |^{2}) | u |^{2} d x \\ ⩾ ‖ u ‖_{ε}^{2} - δ C_{1} ‖ u ‖_{ε}^{2} - C_{δ} ‖ u ‖_{ε}^{2^{}} \\ ⩾ C_{2} ‖ u ‖_{ε}^{2} - C_{δ} ‖ u ‖_{ε}^{2^{}}, \end{array}$ so there exists $r > 0$ independent of u such that $\begin{matrix} (3.14) & ‖ u ‖_{ε} ⩾ r for all u \in N_{ε} . \end{matrix}$

Since f is only continuous, the next results will be fundamental to overcome the non-differentiability of $N_{ε}$ and the incompleteness of $S_{ε}^{+}$ .
Lemma 3.3.
Assume that* $(V_{1})$ – $(V_{2})$ and $(f_{1})$ – $(f_{4})$ hold. Then,
For each $u \in H_{ε}^{+}$ , let $h : R_{+} \to R$ be defined by $h_{u} (t) = J_{ε} (t u)$ . Then, there is a unique $t_{u} > 0$ such that $\begin{array}{l} h_{u}^{'} (t) > 0 in (0, t_{u}), \\ h_{u}^{'} (t) < 0 in (t_{u}, \infty) . \end{array}$

There exists $τ > 0$ independent of u such that $t_{u} ⩾ τ$ for any $u \in S_{ε}^{+}$ . Moreover, for each compact set $K \subset S_{ε}^{+}$ there is a positive constant $C_{K}$ such that $t_{u} ⩽ C_{K}$ for any $u \in K$ .

The map ${\hat{m}}_{ε} : H_{ε}^{+} \to N_{ε}$ given by ${\hat{m}}_{ε} (u) = t_{u} u$ is continuous and $m_{ε} = {\hat{m}}_{ε} |_{S_{ε}^{+}}$ is a homeomorphism between $S_{ε}^{+}$ and $N_{ε}$ . Moreover, $m_{ε}^{- 1} (u) = \frac{u}{‖ u ‖_{ε}}$ .

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

Proof.
(i) It is easy to see that $h_{u} \in C^{1} (R_{+}, R)$ , $h_{u} (0) = 0$ , $h_{u} (t) > 0$ for $t > 0$ small enough and $h_{u} (t) < 0$ for $t > 0$ sufficiently large. Therefore, ${max}_{t ⩾ 0} h_{u} (t)$ is achieved at some $t_{u} > 0$ verifying $h_{u}^{'} (t_{u}) = 0$ and $t_{u} u \in N_{ε}$ . By using $(f_{4})$ and the definition of g, we deduce the uniqueness of a such $t_{u}$ .

(ii) Let $u \in S_{ε}^{+}$ . By (i) there exists $t_{u} > 0$ such that $h_{u}^{'} (t_{u}) = 0$ , or equivalently $\begin{matrix} t_{u} = \int_{R^{N}} g_{ε} (x, | t_{u} u |^{2}) t_{u} | u |^{2} d x . \end{matrix}$ By assumptions $(g_{1})$ and $(g_{2})$ , given $ξ > 0$ there exists a positive constant $C_{ξ}$ such that $\begin{matrix} | g_{ε} (x, t) | ⩽ ξ + C_{ξ} | t |^{\frac{2^{} - 2}{2}}, for every (x, t) \in R^{N} \times R . \end{matrix}$ The above inequality and Sobolev embeddings yield $\begin{array}{l} t_{u} ⩽ ξ t_{u} C_{1} + C_{ξ} C_{2} t_{u}^{2^{} - 1} . \end{array}$ Taking $ξ > 0$ sufficiently small, we obtain that there exists $τ > 0$ , independent of u, such that $t_{u} ⩾ τ$ . Now, let $K \subset S_{ε}^{+}$ be a compact set and we show that $t_{u}$ can be estimated from above by a constant depending on $K$ . Assume by contradiction that there exists a sequence $(u_{n}) \subset K$ such that $t_{n} = t_{u_{n}} \to \infty$ . Therefore, there exists $u \in K$ such that $u_{n} \to u$ in $H_{ε}$ . From (iii) in Lemma 3.1 we get $\begin{matrix} (3.15) & J_{ε} (t_{n} u_{n}) \to - \infty . \end{matrix}$ Fix $v \in N_{ε}$ . Then, using the fact that $⟨ J_{ε}^{'} (v), v ⟩ = 0$ , and assumption $(g_{3})$ , we can infer $\begin{array}{l} J_{ε} (v) & = J_{ε} (v) - \frac{1}{θ} ⟨ J_{ε}^{'} (v), v ⟩ \\ = (\frac{1}{2} - \frac{1}{θ}) ‖ v ‖_{ε}^{2} + \int_{R^{N}} \frac{1}{θ} g_{ε} (x, | v |^{2}) | v |^{2} - \frac{1}{2} G_{ε} (x, | v |^{2})] d x \\ ⩾ (\frac{1}{2} - \frac{1}{θ}) ‖ v ‖_{ε}^{2} + \frac{1}{θ} \int_{Λ_{ε}^{c}} [g_{ε} (x, | v |^{2}) | v |^{2} - \frac{θ}{2} G_{ε} (x, | v |^{2})] d x \\ ⩾ (\frac{θ - 2}{2 θ}) ‖ v ‖_{ε}^{2} - (\frac{θ - 2}{2 θ}) \frac{1}{k} \int_{Λ_{ε}^{c}} V_{ε} (x) | v |^{2} d x \\ (3.16) & ⩾ (\frac{θ - 2}{2 θ}) (1 - \frac{1}{k}) ‖ v ‖_{ε}^{2} . \end{array}$ Taking into account that $θ > 2$ , $k > 1$ , $(t_{u_{n}} u_{n}) \subset N_{ε}$ and $‖ t_{u_{n}} u_{n} ‖_{ε} = t_{u_{n}} \to \infty$ , from (3.16) we deduce that (3.15) does not hold.

(iii) Firstly, we note that ${\hat{m}}_{ε}$ , $m_{ε}$ and $m_{ε}^{- 1}$ are well defined. Indeed, by (i), for each $u \in H_{ε}^{+}$ there exists a unique $m_{ε} (u) \in N_{ε}$ . On the other hand, if $u \in N_{ε}$ then $u \in H_{ε}^{+}$ . Otherwise, if $u \notin H_{ε}^{+}$ , we have $\begin{matrix} | supp (| u |) \cap Λ_{ε} | = 0, \end{matrix}$ which together with $u \in N_{ε}$ and $(g_{3})$ -(ii) gives $\begin{array}{l} 0 < ‖ u ‖_{ε}^{2} & = \int_{R^{N}} g_{ε} (x, | u |^{2}) | u |^{2} d x \\ = \int_{Λ_{ε}} g_{ε} (x, | u |^{2}) | u |^{2} d x + \int_{Λ_{ε}^{c}} g_{ε} (x, | u |^{2}) | u |^{2} d x \\ = \int_{Λ_{ε}^{c}} g_{ε} (x, | u |^{2}) | u |^{2} d x \\ (3.17) & ⩽ \frac{1}{k} \int_{Λ_{ε}^{c}} V_{ε} (x) | u |^{2} d x ⩽ \frac{1}{k} ‖ u ‖_{ε}^{2} . \end{array}$ Using (3.17) we get $\begin{array}{l} 0 < ‖ u ‖_{ε}^{2} ⩽ \frac{1}{k} ‖ u ‖_{ε}^{2}, \end{array}$ and this leads to a contradiction because $k > 1$ . Consequently, $m_{ε}^{- 1} (u) = \frac{u}{‖ u ‖_{ε}} \in S_{ε}^{+}$ , $m_{ε}^{- 1}$ is well defined and continuous. Let $u \in S_{ε}^{+}$ . Then, $\begin{array}{l} m_{ε}^{- 1} (m_{ε} (u)) = m_{ε}^{- 1} (t_{u} u) = \frac{t_{u} u}{‖ t_{u} u ‖_{ε}} = \frac{u}{‖ u ‖_{ε}} = u \end{array}$ from which $m_{ε}$ is a bijection. Now, our aim is to prove that ${\hat{m}}_{ε}$ is a continuous function. Let $(u_{n}) \subset H_{ε}^{+}$ and $u \in H_{ε}^{+}$ such that $u_{n} \to u$ in $H_{ε}^{+}$ . Hence, $\begin{matrix} \frac{u_{n}}{‖ u_{n} ‖_{ε}} \to \frac{u}{‖ u ‖_{ε}} in H_{ε} . \end{matrix}$ Set $v_{n} = \frac{u_{n}}{‖ u_{n} ‖_{ε}}$ and $t_{n} = t_{v_{n}}$ . By (ii) there exists $t_{0} > 0$ such that $t_{n} \to t_{0}$ . Since $t_{n} v_{n} \in N_{ε}$ and $‖ v_{n} ‖_{ε} = 1$ , we have $\begin{matrix} t_{n}^{2} = \int_{R^{N}} g_{ε} (x, | t_{n} v_{n} |^{2}) | t_{n} v_{n} |^{2} d x . \end{matrix}$ Passing to the limit as $n \to \infty$ we obtain $\begin{matrix} t_{0}^{2} = \int_{R^{N}} g_{ε} (x, | t_{0} v |^{2}) | t_{0} v |^{2} d x, \end{matrix}$ where $v = \frac{u}{‖ u ‖_{ε}}$ , that implies that $t_{0} v \in N_{ε}$ . By (i) we deduce that $t_{v} = t_{0}$ , and this shows that $\begin{matrix} {\hat{m}}_{ε} (u_{n}) = {\hat{m}}_{ε} (\frac{u_{n}}{‖ u_{n} ‖_{ε}}) \to {\hat{m}}_{ε} (\frac{u}{‖ u ‖_{ε}}) = {\hat{m}}_{ε} (u) in H_{ε} . \end{matrix}$ Therefore ${\hat{m}}_{ε}$ and $m_{ε}$ are continuous functions.

(iv) Let $(u_{n}) \subset S_{ε}^{+}$ be such that $dist (u_{n}, \partial S_{ε}^{+}) \to 0$ . Then, observing that for each $p \in [2, 2^{}]$ there exists $C_{p} > 0$ such that $\begin{array}{l} ‖ u_{n} ‖_{L^{p} (Λ_{ε})} & ⩽ inf_{v \in \partial S_{ε}^{+}} ‖ u_{n} - v ‖_{L^{p} (Λ_{ε})} \\ ⩽ C_{p} inf_{v \in \partial S_{ε}^{+}} ‖ u_{n} - v ‖_{ε} for all n \in N, \end{array}$ by $(g_{1})$ , $(g_{2})$ , $(g_{3})$ -(ii), $(V_{1})$ - $(V_{2})$ and Sobolev inequality, we can infer $\begin{array}{l} \frac{1}{2} \int_{R^{N}} G_{ε} (x, | t u_{n} |^{2}) d x & = \frac{1}{2} \int_{Λ_{ε}^{c}} G_{ε} (x, | t u_{n} |^{2}) d x + \frac{1}{2} \int_{Λ_{ε}} G_{ε} (x, | t u_{n} |^{2}) d x \\ ⩽ \frac{t^{2}}{2 k} \int_{Λ_{ε}^{c}} V_{ε} (x) | u_{n} |^{2} d x + \int_{Λ_{ε}} \frac{1}{2} F (| t u_{n} |^{2}) + \frac{γ}{2^{}} | t u_{n} |^{2^{}} d x \\ ⩽ \frac{t^{2}}{2 k} ‖ u_{n} ‖_{ε}^{2} + C_{1} t^{2} \int_{Λ_{ε}} | u_{n} |^{2} d x + C_{2} t^{2^{}} \int_{Λ_{ε}} | u_{n} |^{2^{}} d x \\ ⩽ \frac{t^{2}}{2 k} + C_{1}^{'} t^{2} dist {(u_{n}, \partial S_{ε}^{+})}^{2} + C_{2}^{'} t^{2^{}} dist {(u_{n}, \partial S_{ε}^{+})}^{2^{}} \end{array}$ from which, for all $t > 0$ , $\begin{matrix} (3.18) & \frac{1}{2} \underset{n \to \infty}{lim sup} \int_{R^{N}} G_{ε} (x, | t u_{n} |^{2}) d x ⩽ \frac{t^{2}}{2 k} . \end{matrix}$ Bearing in mind the definition of $m_{ε} (u_{n})$ , and by using (3.18), we have $\begin{array}{l} \underset{n \to \infty}{lim inf} J_{ε} (m_{ε} (u_{n})) & ⩾ \underset{n \to \infty}{lim inf} J_{ε} (t u_{n}) \\ = \underset{n \to \infty}{lim inf} [\frac{1}{2} ‖ t u_{n} ‖_{ε}^{2} - \frac{1}{2} \int_{R^{N}} G_{ε} (x, | t u_{n} |^{2}) d x] \\ ⩾ (\frac{1}{2} - \frac{1}{2 k}) t^{2} for all t > 0 . \end{array}$ Recalling that $k > 1$ we get $\begin{matrix} lim_{n \to \infty} J_{ε} (m_{ε} (u_{n})) = \infty . \end{matrix}$ Moreover, by the definition of $J_{ε}$ , we see that $\begin{matrix} \underset{n \to \infty}{lim inf} \frac{1}{2} {‖ m_{ε} (u_{n}) ‖}_{ε}^{2} ⩾ \underset{n \to \infty}{lim inf} J_{ε} (m_{ε} (u_{n})) = \infty . \end{matrix}$ □

Let us define the maps $\begin{matrix} {\hat{ψ}}_{ε} : H_{ε}^{+} \to R and ψ_{ε} : S_{ε}^{+} \to R, \end{matrix}$ by ${\hat{ψ}}_{ε} (u) = J_{ε} ({\hat{m}}_{ε} (u))$ and $ψ_{ε} = {\hat{ψ}}_{ε} |_{S_{ε}^{+}}$ .

The next result is a direct consequence of Lemma 3.3 and Corollary 10 in [42].
Proposition 3.1.
Assume that* $(V_{1})$ – $(V_{2})$ and $(f_{1})$ – $(f_{4})$ hold. Then,
${\hat{ψ}}_{ε} \in C^{1} (H_{ε}^{+}, R)$ and $\begin{matrix} ⟨ {\hat{ψ}}_{ε}^{'} (u), v ⟩ = \frac{‖ {\hat{m}}_{ε} (u) ‖_{ε}}{‖ u ‖_{ε}} ⟨ J_{ε}^{'} ({\hat{m}}_{ε} (u)), v ⟩, \end{matrix}$ for every $u \in H_{ε}^{+}$ and $v \in H_{ε}$ .

$ψ_{ε} \in C^{1} (S_{ε}^{+}, R)$ and $\begin{matrix} ⟨ ψ_{ε}^{'} (u), v ⟩ = {‖ m_{ε} (u) ‖}_{ε} ⟨ J_{ε}^{'} (m_{ε} (u)), v ⟩, \end{matrix}$ for every $v \in T_{u} S_{ε}^{+}$ .

If $(u_{n})$ is a Palais–Smale sequence for $ψ_{ε}$ , then $(m_{ε} (u_{n}))$ is a Palais–Smale sequence for $J_{ε}$ . If $(u_{n}) \subset N_{ε}$ is a bounded Palais–Smale sequence for $J_{ε}$ , then $(m_{ε}^{- 1} (u_{n}))$ is a Palais–Smale sequence for $ψ_{ε}$ .

u is a critical point of $ψ_{ε}$ if and only if $m_{ε} (u)$ is a nontrivial critical point for $J_{ε}$ . Moreover, the corresponding critical values coincide and $\begin{matrix} inf_{u \in S_{ε}^{+}} ψ_{ε} (u) = inf_{u \in N_{ε}} J_{ε} (u) . \end{matrix}$

Remark 3.2.
As in [42], we have the following variational characterization of the infimum of $J_{ε}$ over $N_{ε}$ : $\begin{array}{l} c_{ε} = inf_{u \in N_{ε}} J_{ε} (u) = inf_{u \in H_{ε}^{+}} max_{t > 0} J_{ε} (t u) = inf_{u \in S_{ε}^{+}} max_{t > 0} J_{ε} (t u) . \end{array}$ Moreover, arguing as in [23,39,44], we can verify that $c_{ε} = c_{ε}^{'}$ .
Corollary 3.1.
The functional $ψ_{ε}$ satisfies the ${(P S)}_{c}$ condition on $S_{ε}^{+}$ at any level $c \in R$ if $γ = 0$ , and any level $0 < c < \frac{1}{N} S_{}^{\frac{N}{2}}$ if $γ = 1$ .
Proof.
Let $(u_{n}) \subset S_{ε}^{+}$ be a Palais–Smale sequence for $ψ_{ε}$ at the level c. Then $\begin{matrix} ψ_{ε} (u_{n}) \to c and {‖ ψ_{ε}^{'} (u_{n}) ‖}_{} \to 0, \end{matrix}$ where $‖ \cdot ‖_{}$ is the norm in the dual space ${(T_{u_{n}} S_{ε}^{+})}^{}$ . It follows from Proposition 3.1-(c) that $(m_{ε} (u_{n}))$ is a Palais–Smale sequence for $J_{ε}$ in $H_{ε}$ at the level c. Then, using Lemma 3.2, there exists $u \in S_{ε}^{+}$ such that, up to a subsequence, $\begin{matrix} m_{ε} (u_{n}) \to m_{ε} (u) in H_{ε} . \end{matrix}$ Applying Lemma 3.3-(iii) we can infer that $u_{n} \to u$ in $S_{ε}^{+}$ . □

We end this section by showing the following existence result for (3.2). Theorem 3.1.
Assume that $(V_{1})$ – $(V_{2})$ and $(f_{1})$ – $(f_{4})$ hold. Then, for all $ε > 0$ , there exists a ground state solution to ( 3.2 ).
Proof.
This is a direct consequence of Lemma 3.1, Lemma 3.2, Remark 3.1, the mountain pass theorem [7] and Remark 3.2. □

4. Multiple solutions for the modified problem

4.1. The limiting scalar problem

In this section we prove a multiplicity result for problem (3.2). We start by considering the limiting scalar problem associated with (3.1), namely $\begin{matrix} (4.1) & \{\begin{matrix} - Δ u + V_{0} u = f (u^{2}) u + γ {(u^{+})}^{2^{*} - 1} in R^{N}, \\ u \in H^{1} (R^{N}, R), u > 0 in R^{N}, \end{matrix} \end{matrix}$ and we introduce the corresponding energy functional $I_{0} : H_{0} \to R$ given by $\begin{array}{l} I_{0} (u) = \frac{1}{2} ‖ u ‖_{0}^{2} - \int_{R^{N}} [\frac{1}{2} F (u^{2}) + \frac{γ}{2^{*}} {(u^{+})}^{2^{*}}] d x, \end{array}$ where $H_{0}$ stands for the Sobolev space $H^{1} (R^{N}, R)$ endowed with the norm $\begin{matrix} ‖ u ‖_{0} = {(‖ \nabla u ‖_{L^{2} (R^{N})}^{2} + V_{0} ‖ u ‖_{L^{2} (R^{N})}^{2})}^{\frac{1}{2}} . \end{matrix}$ For any $u, v \in H_{0}$ , we set $\begin{matrix} {⟨ u, v ⟩}_{0} = \int_{R^{N}} \nabla u \cdot \nabla v d x + V_{0} \int_{R^{N}} u v d x . \end{matrix}$ Let $\begin{matrix} M_{0} = {u \in H_{0} ∖ {0} : ⟨ I_{0}^{'} (u), u ⟩ = 0} \end{matrix}$ the Nehari manifold associated with $I_{0}$ . Let us consider $\begin{matrix} H_{0}^{+} = {u \in H_{0} : | supp (| u |) | > 0} \subset H_{ε} . \end{matrix}$ Let $S_{0}$ be the unit sphere of $H_{0}$ and we denote by $S_{0}^{+} = S_{0} \cap H_{0}^{+}$ . We observe that $H_{0}^{+}$ is open in $H_{0}$ . By the definition of $S_{0}^{+}$ and the fact that $H_{0}^{+}$ is open in $H_{0}$ , it follows that $S_{0}^{+}$ is a incomplete $C^{1, 1}$ -manifold of codimension 1, modeled on $H_{0}$ and contained in the open $H_{0}^{+}$ . Then, $H_{0} = T_{u} S_{0}^{+} \oplus R u$ for each $u \in S_{0}^{+}$ , where $\begin{matrix} T_{u} S_{0}^{+} = {v \in H_{0} : {⟨ u, v ⟩}_{0} = 0} . \end{matrix}$ Arguing as in the proof of Lemma 3.3 we have the following result.

Lemma 4.1.
Assume that $(f_{1})$ – $(f_{4})$ hold. Then,
For each $u \in H_{0}^{+}$ , let $h : R_{+} \to R$ be defined by $h_{u} (t) = I_{0} (t u)$ . Then, there is a unique $t_{u} > 0$ such that $\begin{array}{l} h_{u}^{'} (t) > 0 in (0, t_{u}), \\ h_{u}^{'} (t) < 0 in (t_{u}, \infty) . \end{array}$

There exists $τ > 0$ independent of u such that $t_{u} ⩾ τ$ for any $u \in S_{0}^{+}$ . Moreover, for each compact set $K \subset S_{0}^{+}$ there is a positive constant $C_{K}$ such that $t_{u} ⩽ C_{K}$ for any $u \in K$ .

The map ${\hat{m}}_{0} : H_{0}^{+} \to M_{0}$ given by ${\hat{m}}_{0} (u) = t_{u} u$ is continuous and $m_{0} = {\hat{m}}_{0} |_{S_{0}^{+}}$ is a homeomorphism between $S_{0}^{+}$ and $N_{0}$ . Moreover, $m_{0}^{- 1} (u) = \frac{u}{‖ u ‖_{0}}$ .

If there is a sequence $(u_{n}) \subset S_{0}^{+}$ such that $dist (u_{n}, \partial S_{0}^{+}) \to 0$ then $‖ m_{0} (u_{n}) ‖_{0} \to \infty$ and $I_{0} (m_{0} (u_{n})) \to \infty$ .

Let us define the maps $\begin{matrix} {\hat{ψ}}_{0} : H_{0}^{+} \to R and ψ_{0} : S_{0}^{+} \to R, \end{matrix}$ by ${\hat{ψ}}_{0} (u) = I_{0} ({\hat{m}}_{0} (u))$ and $ψ_{0} = {\hat{ψ}}_{0} |_{S_{0}^{+}}$ .

The next result is a direct consequence of Lemma 4.1 and Corollary 10 in [42].
Proposition 4.1.
Assume that $(f_{1})$ – $(f_{4})$ hold. Then,
${\hat{ψ}}_{0} \in C^{1} (H_{0}^{+}, R)$ and $\begin{matrix} ⟨ {\hat{ψ}}_{0}^{'} (u), v ⟩ = \frac{‖ {\hat{m}}_{0} (u) ‖_{0}}{‖ u ‖_{0}} ⟨ I_{0}^{'} ({\hat{m}}_{0} (u)), v ⟩, \end{matrix}$ for every $u \in H_{0}^{+}$ and $v \in H_{0}$ .

$ψ_{0} \in C^{1} (S_{0}^{+}, R)$ and $\begin{matrix} ⟨ ψ_{0}^{'} (u), v ⟩ = {‖ m_{0} (u) ‖}_{0} ⟨ I_{0}^{'} (m_{0} (u)), v ⟩, \end{matrix}$ for every $v \in T_{u} S_{0}^{+}$ .

If $(u_{n})$ is a Palais–Smale sequence for $ψ_{0}$ , then $(m_{0} (u_{n}))$ is a Palais–Smale sequence for $I_{0}$ . If $(u_{n}) \subset M_{0}$ is a bounded Palais–Smale sequence for $I_{0}$ , then $(m_{0}^{- 1} (u_{n}))$ is a Palais–Smale sequence for $ψ_{0}$ .

u is a critical point of $ψ_{0}$ if and only if $m_{0} (u)$ is a nontrivial critical point for $I_{0}$ . Moreover, the corresponding critical values coincide and $\begin{matrix} inf_{u \in S_{0}^{+}} ψ_{0} (u) = inf_{u \in M_{0}} I_{0} (u) . \end{matrix}$

Remark 4.1.
As in [42], we have the following variational characterization of the infimum of $I_{0}$ over $M_{0}$ : $\begin{array}{l} d_{0} & = inf_{u \in M_{0}} I_{0} (u) = inf_{u \in H_{0}^{+}} max_{t > 0} I_{0} (t u) = inf_{u \in S_{0}^{+}} max_{t > 0} I_{0} (t u) . \end{array}$ We also recall that, when $γ = 1$ , $0 < d_{0} < \frac{1}{N} S_{}^{\frac{N}{2}}$ ; see [2].

The next lemma allows us to assume that the weak limit of a Palais–Smale sequence at the level $d_{0}$ is nontrivial.
Lemma 4.2.
Let* $(u_{n}) \subset H_{0}$ be a Palais–Smale sequence for $I_{0}$ at the level $d_{0}$ . Then
either $u_{n} \to 0$ in $H_{0}$ , or

there exist a sequence $(y_{n}) \subset R^{N}$ and constants $R, β > 0$ such that $\begin{matrix} \underset{n \to \infty}{lim inf} \int_{B_{R} (y_{n})} | u_{n} |^{2} d x ⩾ β > 0 . \end{matrix}$

Proof.
Assume that (ii) does not occur. Arguing as in the proof of Step 1 in Lemma 3.2, we see that $(u_{n})$ is bounded in $H_{0}$ . Then we use Lemma 2.1 to deduce that $u_{n} \to 0$ in $L^{r} (R^{N}, R)$ for all $r \in (2, 2^{})$ . In view of $(f_{1})$ – $(f_{2})$ , we get $\int_{R^{N}} f (u_{n}^{2}) u_{n}^{2} d x = o_{n} (1)$ . When $γ = 0$ , the previous fact combined with $⟨ I_{0}^{'} (u_{n}), u_{n} ⟩ = o_{n} (1)$ yields $‖ u_{n} ‖_{0}^{2} = o_{n} (1)$ . When $γ = 1$ , then we can see that $‖ u_{n} ‖_{0}^{2} = ‖ u_{n}^{+} ‖_{L^{2^{}} (R^{N})}^{2^{}} + o_{n} (1)$ . Since $(u_{n})$ is bounded in $H_{0}$ , we may assume that $‖ u_{n} ‖_{0}^{2} \to ℓ$ and $‖ u_{n}^{+} ‖_{L^{2^{}} (R^{N})}^{2^{}} \to ℓ$ for some $ℓ > 0$ . Assume by contradiction that $ℓ > 0$ . Then, using $I_{0} (u_{n}) = d_{0} + o_{n} (1)$ , we see that $ℓ = N d_{0}$ . On the other hand, by Sobolev inequality, we have $‖ \nabla u_{n} ‖_{L^{2} (R^{N})}^{2} ⩾ S_{} {(‖ u_{n}^{+} ‖_{L^{2^{}} (R^{N})}^{2^{}})}^{\frac{2}{2^{}}}$ . Then, taking the limit as $n \to \infty$ , we find $ℓ ⩾ S_{}^{\frac{N}{2}}$ which combined with $ℓ = N d_{0}$ gives $d_{0} ⩾ \frac{1}{N} S_{}^{\frac{N}{2}}$ which is a contradiction. □
Remark 4.2.
Let us observe that, if u is the weak limit of a Palais–Smale sequence $(u_{n})$ of $I_{0}$ at the level $d_{0}$ , then we can assume that $u \neq 0$ . Otherwise, we would have $u_{n} ⇀ 0$ in $H_{0}$ and, if $u_{n} ↛ 0$ in $H_{0}$ , we conclude from Lemma 4.2 that there are $(y_{n}) \subset R^{N}$ and $R, β > 0$ such that $\begin{matrix} \underset{n \to \infty}{lim inf} \int_{B_{R} (y_{n})} | u_{n} |^{2} d x ⩾ β > 0 . \end{matrix}$ Set $v_{n} (x) = u_{n} (x + y_{n})$ . Then we see that $(v_{n})$ is a Palais–Smale sequence for $I_{0}$ at the level $d_{0}$ , $(v_{n})$ is bounded in $H_{0}$ and there exists $v \in H_{0}$ such that $v_{n} ⇀ v$ with $v \neq 0$ .

Next we prove an existence result for the autonomous problem.
Theorem 4.1.
There exists a positive ground state solution of (* 4.1 ).
Proof.
Let $(u_{n}) \subset H_{0}$ be a Palais–Smale sequence for $I_{0}$ at the level $d_{0}$ . Arguing as in the proof of Step 1 in Lemma 3.2, we can see that $(u_{n})$ is bounded in $H_{0}$ , so we may assume that $u_{n} ⇀ u$ in $H_{0}$ . Standard arguments show that u is a critical point of $I_{0}$ . Using Lemma 4.2 and Remark 4.2, we may suppose that u is not trivial. Hence $u \in M_{0}$ . Now we prove that $I_{0} (u) = d_{0}$ . Indeed, thanks to $u \in M_{0}$ , $(f_{3})$ and Fatou’s Lemma, we have $\begin{array}{l} d_{0} & ⩽ I_{0} (u) - \frac{1}{θ} ⟨ I_{0}^{'} (u), u ⟩ \\ ⩽ \underset{n \to \infty}{lim inf} [\frac{1}{2} ‖ u_{n} ‖_{0}^{2} + \int_{R^{N}} \frac{1}{θ} [f (u_{n}^{2}) u_{n}^{2} + γ {(u_{n}^{+})}^{2^{}}] - [\frac{1}{2} F (u_{n}^{2}) + \frac{γ}{2^{}} {(u_{n}^{+})}^{2^{}}] d x] \\ = \underset{n \to \infty}{lim inf} [I_{0} (u_{n}) - \frac{1}{θ} ⟨ I_{0}^{'} (u_{n}), u_{n} ⟩] \\ = d_{0} . \end{array}$ In view of $⟨ I_{0}^{'} (u), u^{-} ⟩ = 0$ and $(f_{1})$ we can see that $u ⩾ 0$ in $R^{N}$ . Standard arguments (see [33]) show that $u \in L^{\infty} (R^{N}, R) \cap C_{loc}^{1, α} (R^{N}, R)$ for some $α \in (0, 1)$ , and by using the Harnack inequality [29] we deduce that $u > 0$ in $R^{N}$ . □

The next lemma is a compactness result for the autonomous problem which will be used later.
Lemma 4.3.
Let* $(u_{n}) \subset M_{0}$ be a sequence such that $I_{0} (u_{n}) \to d_{0}$ . Then $(u_{n})$ has a convergent subsequence in $H_{0}$ .
Proof.
Since $(u_{n}) \subset M_{0}$ and $I_{0} (u_{n}) \to d_{0}$ , we can apply Lemma 4.1-(iii), Proposition 4.1-(d) and the definition of $d_{0}$ to infer that $\begin{matrix} v_{n} = m^{- 1} (u_{n}) = \frac{u_{n}}{‖ u_{n} ‖_{0}} \in S_{0}^{+} \end{matrix}$ and $\begin{matrix} ψ_{0} (v_{n}) = I_{0} (u_{n}) \to d_{0} = inf_{v \in S_{0}^{+}} ψ_{0} (v) . \end{matrix}$ Let us introduce the following map $F : {\overline{S}}_{0}^{+} \to R \cup {\infty}$ defined by setting $\begin{matrix} F (u) = \{\begin{matrix} ψ_{0} (u) & if u \in S_{0}^{+}, \\ \infty & if u \in \partial S_{0}^{+} . \end{matrix} \end{matrix}$ We note that
$({\overline{S}}_{0}^{+}, d_{0})$ , where $d (u, v) = ‖ u - v ‖_{0}$ , is a complete metric space;

$F \in C ({\overline{S}}_{0}^{+}, R \cup {\infty})$ , by Lemma 4.1-(iv);

$F$ is bounded below, by Proposition 4.1-(d).
By applying Ekeland’s variational principle [44], we can find $({\hat{v}}_{n}) \subset S_{0}^{+}$ such that $({\hat{v}}_{n})$ is a Palais–Smale sequence for $ψ_{0}$ at the level $d_{0}$ on $S_{0}^{+}$ and $‖ {\hat{v}}_{n} - v_{n} ‖_{0} = o_{n} (1)$ . Now the remainder of the proof follows from Proposition 4.1, Theorem 4.1 and arguing as in the proof of Corollary 3.1. □

Arguing as in the proof of Lemma 3.4 in [8], we have the following interesting relation between $c_{ε}$ and $d_{0}$ . Lemma 4.4.
The numbers $c_{ε}$ and $d_{0}$ satisfy the following inequality: $\begin{matrix} \underset{ε \to 0}{lim sup} c_{ε} ⩽ d_{0} . \end{matrix}$

4.2. Technical results

In this section we make use of the Lusternik–Schnirelman category theory to obtain multiple solutions to (3.2). In particular, we relate the number of positive solutions of (3.2) to the topology of the set M. For this reason, we take $δ > 0$ such that $\begin{matrix} M_{δ} = {x \in R^{N} : dist (x, M) ⩽ δ} \subset Λ, \end{matrix}$ and consider a smooth nonicreasing function $η : [0, \infty) \to R$ such that $η (t) = 1$ if $0 ⩽ t ⩽ \frac{δ}{2}$ , $η (t) = 0$ if $t ⩾ δ$ , $0 ⩽ η ⩽ 1$ and $| η^{'} (t) | ⩽ c$ for some $c > 0$ .

For any $y \in M$ , we introduce $\begin{matrix} Ψ_{ε, y} (x) = η (| ε x - y |) w (\frac{ε x - y}{ε}) e^{ı τ_{y} (\frac{ε x - y}{ε})}, \end{matrix}$ where $τ_{y} (x) = \sum_{j = 1}^{N} A_{j} (y) x_{j}$ and $w \in H^{1} (R^{N}, R)$ is a positive ground state solution to the autonomous problem (4.1) whose existence is guaranteed by Theorem 4.1. Let $t_{ε} > 0$ be the unique number such that $\begin{matrix} J_{ε} (t_{ε} Ψ_{ε, y}) = max_{t ⩾ 0} J_{ε} (t Ψ_{ε, y}) . \end{matrix}$ Finally, we consider $Φ_{ε} : M \to N_{ε}$ defined by setting $\begin{matrix} Φ_{ε} (y) = t_{ε} Ψ_{ε, y} . \end{matrix}$

Lemma 4.5.
The functional $Φ_{ε}$ satisfies the following limit $\begin{matrix} lim_{ε \to 0} J_{ε} (Φ_{ε} (y)) = d_{0} uniformly in y \in M . \end{matrix}$
Proof.
Assume by contradiction that there exist $δ_{0} > 0$ , $(y_{n}) \subset M$ and $ε_{n} \to 0$ such that $\begin{matrix} (4.2) & | J_{ε_{n}} (Φ_{ε_{n}} (y_{n})) - d_{0} | ⩾ δ_{0} . \end{matrix}$ Applying Lemma 3.2 in [21] and the dominated convergence theorem, we know that $\begin{array}{l} (4.3) & ‖ Ψ_{ε_{n}, y_{n}} ‖_{ε_{n}}^{2} \to ‖ w ‖_{0}^{2} \in (0, \infty) . \end{array}$ On the other hand, since $⟨ J_{ε_{n}}^{'} (Φ_{ε_{n}} (y_{n})), Φ_{ε_{n}} (y_{n}) ⟩ = 0$ and using the change of variable $z = \frac{ε_{n} x - y_{n}}{ε_{n}}$ , we have that $\begin{array}{l} t_{ε_{n}}^{2} ‖ Ψ_{ε_{n}, y_{n}} ‖_{ε_{n}}^{2} = \int_{R^{N}} g (ε_{n} z + y_{n}, {| t_{ε_{n}} η (| ε_{n} z |) w (z) |}^{2}) {| t_{ε_{n}} η (| ε_{n} z |) w (z) |}^{2} d z . \end{array}$ If $z \in B_{\frac{δ}{ε_{n}}}$ , then $ε_{n} z + y_{n} \in B_{δ} (y_{n}) \subset M_{δ} \subset Λ$ . Since $g (x, t) = f (t) + γ t^{\frac{2^{} - 2}{2}}$ for $(x, t) \in Λ \times [0, \infty)$ , we get $\begin{array}{l} (4.4) & t_{ε_{n}}^{2} ‖ Ψ_{ε_{n}, y_{n}} ‖_{ε_{n}}^{2} = \int_{R^{N}} f ({| t_{ε_{n}} η (| ε_{n} z |) w (z) |}^{2}) {(t_{ε_{n}} η (| ε_{n} z |) w (z))}^{2} + γ {(t_{ε_{n}} η (| ε_{n} z |) w (z))}^{2^{}} d z . \end{array}$ In view of $η (| x |) = 1$ for $x \in B_{\frac{δ}{2}}$ and that $B_{\frac{δ}{2}} \subset B_{\frac{δ}{2 ε_{n}}}$ for all n large enough, it follows from (4.4) and $(f_{4})$ that $\begin{array}{l} ‖ Ψ_{ε_{n}, y_{n}} ‖_{ε_{n}}^{2} & ⩾ \int_{B_{\frac{δ}{2}}} \frac{f (| t_{ε_{n}} w (z) |^{2}) {(t_{ε_{n}} w (z))}^{2} + γ {(t_{ε_{n}} w (z))}^{2^{}}}{t_{ε_{n}}^{2}} d z \\ (4.5) & ⩾ f ({| t_{ε_{n}} w (\hat{z}) |}^{2}) \int_{B_{\frac{δ}{2}}} w^{2} (z) d z, \end{array}$ where $w (\hat{z}) = {min}_{z \in {\overline{B}}_{\frac{δ}{2}}} w (z) > 0$ (we recall that $w \in C (R^{N}, R)$ and $w > 0$ in $R^{N}$ ). Now, assume by contradiction that $t_{ε_{n}} \to \infty$ . This fact, (4.3) and (4.5) yield $\begin{matrix} ‖ w ‖_{0}^{2} = \infty, \end{matrix}$ that is a contradiction. Hence, $(t_{ε_{n}})$ is bounded and, up to subsequence, we may assume that $t_{ε_{n}} \to t_{0}$ for some $t_{0} ⩾ 0$ . In particular, $t_{0} > 0$ . In fact, if $t_{0} = 0$ , we see that (3.14) and (4.4) imply that $\begin{matrix} r ⩽ \int_{R^{3}} f ({| t_{ε_{n}} η (| ε_{n} z |) w (z) |}^{2}) {(t_{ε_{n}} η (| ε_{n} z |) w (z))}^{2} + γ {(t_{ε_{n}} η (| ε_{n} z |) w (z))}^{2^{}} d z . \end{matrix}$ Using $(f_{1})$ , $(f_{2})$ , (4.3) and the above inequality, we get a contradiction. Hence, $t_{0} > 0$ . Thus, letting $n \to \infty$ in (4.4), we have that $\begin{array}{l} t_{0}^{2} ‖ w ‖_{0}^{2} = \int_{R^{N}} f ({(t_{0} w)}^{2}) {(t_{0} w)}^{2} + γ {(t_{0} w)}^{2^{}} d x . \end{array}$ Bearing in mind that $w \in M_{0}$ and using $(f_{4})$ , we infer that $t_{0} = 1$ . Passing to the limit as $n \to \infty$ and using $t_{ε_{n}} \to 1$ we conclude that $\begin{matrix} lim_{n \to \infty} J_{ε_{n}} (Φ_{ε_{n}, y_{n}}) = I_{0} (w) = d_{0}, \end{matrix}$ which is in contrast with (4.2). □

Let us fix $ρ = ρ (δ) > 0$ satisfying $M_{δ} \subset B_{ρ}$ , and we consider $Υ : R^{N} \to R^{N}$ given by $\begin{matrix} Υ (x) = \{\begin{matrix} x & if | x | < ρ, \\ \frac{ρ x}{| x |} & if | x | ⩾ ρ . \end{matrix} \end{matrix}$ Then we define the barycenter map $β_{ε} : N_{ε} \to R^{N}$ as follows $\begin{array}{l} β_{ε} (u) = \frac{\int_{R^{N}} Υ (ε x) | u (x) |^{2} d x}{\int_{R^{N}} | u (x) |^{2} d x} . \end{array}$ Since $M \subset B_{ρ}$ , the definition of Υ and the dominated convergence theorem imply that $\begin{array}{l} (4.6) & lim_{ε \to 0} β_{ε} (Φ_{ε} (y)) = y uniformly in y \in M . \end{array}$ The next compactness result will play a fundamental role to prove that the solutions of (3.2) are also solution to (3.1).
Lemma 4.6.
Let* $ε_{n} \to 0$ and $(u_{n}) \subset N_{ε_{n}}$ be such that $J_{ε_{n}} (u_{n}) \to d_{0}$ . Then there exists $({\tilde{y}}_{n}) \subset R^{N}$ such that $v_{n} (x) = | u_{n} | (x + {\tilde{y}}_{n})$ has a convergent subsequence in $H_{0}$ . Moreover, up to a subsequence, $y_{n} = ε_{n} {\tilde{y}}_{n} \to y_{0}$ for some $y_{0} \in M$ .
Proof.
In the case $γ = 0$ , it is enough to argue as in the proof of Lemmas 3.4 in [6]. We only have to use Lemma 4.3 instead of Lemmas 3.1 in [6] to obtain the strong convergence of $(v_{n})$ in $H_{0}$ . In the case $γ = 1$ , the only main difference consists in proving the formula $(3.15)$ in [6], that is there exist $({\tilde{y}}_{n}) \subset R^{N}$ and constants $R, β > 0$ such that $\begin{array}{l} (4.7) & \underset{n \to \infty}{lim sup} \int_{B_{R} ({\tilde{y}}_{n})} | u_{n} |^{2} d x ⩾ β . \end{array}$ Suppose by contradiction that condition (4.7) does not hold. Then, for all $R > 0$ , we have $\begin{matrix} lim_{n \to \infty} sup_{y \in R^{N}} \int_{B_{R} (y)} | u_{n} |^{2} d x = 0 . \end{matrix}$ Since we know that $(| u_{n} |)$ is bounded in $H^{1} (R^{N}, R)$ , we can use Lemma 2.1 to deduce that $| u_{n} | \to 0$ in $L^{r} (R^{N}, R)$ for any $r \in (2, 2^{})$ . By $(f_{1})$ and $(f_{2})$ , it follows that $\begin{array}{l} \int_{R^{N}} F (| u_{n} |^{2}) d x = \int_{R^{N}} f (| u_{n} |^{2}) | u_{n} |^{2} d x = o_{n} (1) . \end{array}$ This implies that $\begin{array}{l} (4.8) & \frac{1}{2} \int_{R^{N}} G_{ε_{n}} (x, | u_{n} |^{2}) d x ⩽ \frac{1}{2^{}} \int_{Λ_{ε_{n}} \cup {| u_{n} |^{2} ⩽ a}} | u_{n} |^{2^{}} d x + \frac{V_{0}}{2 k} \int_{Λ_{ε_{n}}^{c} \cap {| u_{n} |^{2} > a}} | u_{n} |^{2} d x + o_{n} (1) \end{array}$ and $\begin{array}{l} (4.9) & \int_{R^{N}} g_{ε_{n}} (x, | u_{n} |^{2}) | u_{ε_{n}} |^{2} d x = \int_{Λ_{ε_{n}} \cup {| u_{n} |^{2} ⩽ a}} | u_{n} |^{2^{}} d x + \frac{V_{0}}{k} \int_{Λ_{ε_{n}}^{c} \cap {| u_{n} |^{2} > a}} | u_{n} |^{2} d x + o_{n} (1) . \end{array}$ Taking into account $⟨ J_{ε_{n}}^{'} (u_{n}), u_{n} ⟩ = 0$ and (4.9), we can deduce that $\begin{array}{l} (4.10) & ‖ u_{n} ‖_{ε_{n}}^{2} - \frac{V_{0}}{k} \int_{Λ_{ε_{n}}^{c} \cap {| u_{n} |^{2} > a}} | u_{n} |^{2} d x = \int_{Λ_{ε_{n}} \cup {| u_{n} |^{2} ⩽ a}} | u_{n} |^{2^{}} d x + o_{n} (1) . \end{array}$ Let $ℓ ⩾ 0$ be such that $\begin{matrix} ‖ u_{n} ‖_{ε_{n}}^{2} - \frac{V_{0}}{k} \int_{Λ_{ε_{n}}^{c} \cap {| u_{n} |^{2} > a}} | u_{n} |^{2} d x \to ℓ . \end{matrix}$ It is easy to see that $ℓ > 0$ , otherwise $u_{n} \to 0$ in $H_{ε_{n}}$ and thus $J_{ε_{n}} (u_{ε_{n}}) \to 0$ in contrast with $d_{0} > 0$ . It follows from (4.10) that $\begin{matrix} \int_{Λ_{ε_{n}} \cup {| u_{n} |^{2} ⩽ a}} | u_{n} |^{2^{}} d x \to ℓ . \end{matrix}$ Using $J_{ε_{n}} (u_{n}) - \frac{1}{2^{}} ⟨ J_{ε}^{'} (u_{n}), u_{n} ⟩ = d_{0} + o_{n} (1)$ , (4.8), (4.9) and (4.10), we can see that $\begin{array}{l} (4.11) & \frac{1}{N} ℓ ⩽ d_{0} . \end{array}$ Now, from the definition of $S_{}$ , we obtain that $\begin{matrix} ‖ u_{n} ‖_{ε_{n}}^{2} - \frac{V_{0}}{k} \int_{Λ_{ε_{n}}^{c} \cap {| u_{n} |^{2} > a}} | u_{n} |^{2} d x ⩾ S_{} {(\int_{Λ_{ε_{n}} \cup {| u_{n} |^{2} ⩽ a}} | u_{n} |^{2^{}} d x)}^{\frac{2}{2^{}}}, \end{matrix}$ and taking the limit as $n \to \infty$ we can infer that $ℓ ⩾ S_{}^{\frac{N}{2}}$ . This fact combined with (4.11) yields $\begin{matrix} d_{0} ⩾ \frac{1}{N} S_{}^{\frac{N}{2}} \end{matrix}$ which contradicts Remark 4.1. This completes the proof of (4.7). □

Now, we consider the following subset of $N_{ε}$ : $\begin{matrix} {\tilde{N}}_{ε} = {u \in N_{ε} : J_{ε} (u) ⩽ d_{0} + h (ε)}, \end{matrix}$ where $h (ε) = {sup}_{y \in M} | J_{ε} (Φ_{ε} (y)) - d_{0} | \to 0$ as $ε \to 0$ as a consequence of Lemma 4.5. By the definition of $h (ε)$ , we know that, for all $y \in M$ and $ε > 0$ , $Φ_{ε} (y) \in {\tilde{N}}_{ε}$ and ${\tilde{N}}_{ε} \neq \emptyset$ . Arguing as in the proof of Proposition 3.3 in [6], we have:
Lemma 4.7.
For any* $δ > 0$ , there holds that $\begin{matrix} lim_{ε \to 0} sup_{u \in {\tilde{N}}_{ε}} dist (β_{ε} (u), M_{δ}) = 0 . \end{matrix}$

We end this section by proving a multiplicity result for (3.2). Since $S_{ε}^{+}$ is not a completed metric space, we cannot use directly an abstract result as in [3,4,6]. Instead, we invoke the abstract category result in [42]. Theorem 4.2.
For any $δ > 0$ such that $M_{δ} \subset Λ$ , there exists ${\tilde{ε}}_{δ} > 0$ such that, for any $ε \in (0, {\tilde{ε}}_{δ})$ , problem ( 3.2 ) has at least ${cat}_{M_{δ}} (M)$ nontrivial solutions.
Proof.
For any $ε > 0$ , we consider the map $α_{ε} : M \to S_{ε}^{+}$ defined as $α_{ε} (y) = m_{ε}^{- 1} (Φ_{ε} (y))$ .

Using Lemma 4.5, we see that $\begin{matrix} (4.12) & lim_{ε \to 0} ψ_{ε} (α_{ε} (y)) = lim_{ε \to 0} J_{ε} (Φ_{ε} (y)) = d_{0} uniformly in y \in M . \end{matrix}$ Set $\begin{matrix} {\tilde{S}}_{ε}^{+} = {w \in S_{ε}^{+} : ψ_{ε} (w) ⩽ d_{0} + h (ε)}, \end{matrix}$ where $h (ε) = {sup}_{y \in M} | ψ_{ε} (α_{ε} (y)) - d_{0} |$ . It follows from (4.12) that $h (ε) \to 0$ as $ε \to 0$ . Moreover, $α_{ε} (y) \in {\tilde{S}}_{ε}^{+}$ for all $y \in M$ and this shows that ${\tilde{S}}_{ε}^{+} \neq \emptyset$ for all $ε > 0$ .

From the above considerations, together with Lemma 4.5, Lemma 3.3-(iii), (4.6) and Lemma 4.7, we can find $\bar{ε} = {\bar{ε}}_{δ} > 0$ such that the following diagram $\begin{matrix} M \overset{Φ_{ε}}{\to} Φ_{ε} (M) \overset{m_{ε}^{- 1}}{\to} α_{ε} (M) \overset{m_{ε}}{\to} Φ_{ε} (M) \overset{β_{ε}}{\to} M_{δ} \end{matrix}$ is well defined for any $ε \in (0, \bar{ε})$ . Thanks to (4.6), and decreasing $\bar{ε}$ if necessary, we see that $β_{ε} (Φ_{ε} (y)) = y + θ (ε, y)$ for all $y \in M$ , for some function $θ (ε, y)$ satisfying $| θ (ε, y) | < \frac{δ}{2}$ uniformly in $y \in M$ and for all $ε \in (0, \bar{ε})$ . Define $H (t, y) = y + (1 - t) θ (ε, y)$ . Then $H : [0, 1] \times M \to M_{δ}$ is continuous. Clearly, $H (0, y) = β_{ε} (Φ_{ε} (y))$ and $H (1, y) = y$ for all $y \in M$ . Consequently, $H (t, y)$ is a homotopy between $β_{ε} \circ Φ_{ε} = (β_{ε} \circ m_{ε}) \circ (m_{ε}^{- 1} \circ Φ_{ε})$ and the inclusion map $id : M \to M_{δ}$ . This fact yields $\begin{matrix} (4.13) & {cat}_{α_{ε} (M)} α_{ε} (M) ⩾ {cat}_{M_{δ}} (M) . \end{matrix}$ Applying Corollary 3.1, Lemma 4.4, and Theorem 27 in [42] with $c = c_{ε} ⩽ d_{0} + h (ε) = d$ and $K = α_{ε} (M)$ , we obtain that $ψ_{ε}$ has at least ${cat}_{α_{ε} (M)} α_{ε} (M)$ critical points on ${\tilde{S}}_{ε}^{+}$ . Taking into account Proposition 3.1-(d) and (4.13), we infer that $J_{ε}$ admits at least ${cat}_{M_{δ}} (M)$ critical points in ${\tilde{N}}_{ε}$ . □

5. Proof of Theorem 1.1

This last section is devoted to the proof of the first main result of this paper. In order to show that the solutions of (3.2) are indeed solutions to (3.1) for $ε > 0$ small, we need to verify that (3.3) holds true provided that $ε > 0$ is sufficiently small. Now we prove the following crucial result.

Lemma 5.1.
Let $ε_{n} \to 0$ and $(u_{n}) \subset {\tilde{N}}_{ε_{n}}$ be a sequence of solutions to ( 3.2 ). Then, $v_{n} = | u_{n} | (\cdot + {\tilde{y}}_{n})$ satisfies $v_{n} \in L^{\infty} (R^{N}, R)$ and there exists $C > 0$ such that $\begin{matrix} (5.1) & ‖ v_{n} ‖_{L^{\infty} (R^{N})} ⩽ C for all n \in N, \end{matrix}$ where $({\tilde{y}}_{n})$ is given by Lemma 4.6 . Moreover, $\begin{array}{l} (5.2) & lim_{| x | \to \infty} v_{n} (x) = 0 uniformly in n \in N . \end{array}$
Proof.
The proof of this result can be obtained arguing as in the proof of Lemma 4.1 in [6]. Anyway, we prefer to give an alternative proof. Since $J_{ε_{n}} (u_{n}) ⩽ d_{0} + h (ε_{n})$ with $h (ε_{n}) \to 0$ as $n \to \infty$ , we can argue as at the beginning of the proof of Lemma 4.6 to deduce that $J_{ε_{n}} (u_{n}) \to d_{0}$ . Thus we may invoke Lemma 4.6 to obtain a sequence $({\tilde{y}}_{n}) \subset R^{N}$ such that $ε_{n} {\tilde{y}}_{n} \to y_{0} \in M$ and $v_{n} = | u_{n} | (\cdot + {\tilde{y}}_{n})$ strongly converges in $H_{0}$ . Let ${\tilde{u}}_{n} (x) = u_{n} (\cdot + {\tilde{y}}_{n})$ and note that it solves $\begin{matrix} (5.3) & {(\frac{1}{ı} \nabla - {\tilde{A}}_{n} (x))}^{2} {\tilde{u}}_{n} + {\tilde{V}}_{n} (x) {\tilde{u}}_{n} = {\tilde{g}}_{n} (x, v_{n}^{2}) {\tilde{u}}_{n} in R^{N}, \end{matrix}$ where $\begin{array}{l} {\tilde{A}}_{n} (x) = A_{ε_{n}} (x + {\tilde{y}}_{n}), \\ {\tilde{V}}_{n} (x) = V_{ε_{n}} (x + {\tilde{y}}_{n}), \end{array}$ and $\begin{matrix} {\tilde{g}}_{n} (x, v_{n}^{2}) = g (ε_{n} x + ε_{n} {\tilde{y}}_{n}, v_{n}^{2}) . \end{matrix}$ Using Kato’s inequality (2.2), we deduce that $v_{n}$ satisfies (in distributional sense) $\begin{matrix} (5.4) & - Δ v_{n} + {\tilde{V}}_{n} (x) v_{n} ⩽ {\tilde{g}}_{n} (x, v_{n}^{2}) v_{n} = h_{n} in R^{N} . \end{matrix}$ Then, using the strong convergence of $(v_{n})$ in $H_{0}$ , we can perform a Moser iteration argument [37] as in Lemma 4.5 in [4] to obtain the assertion. □

Now, we are ready to give the proof of Theorem 1.1. Proof of Theorem 1.1.
Let $δ > 0$ be such that $M_{δ} \subset Λ$ . Firstly, we claim that there exists ${\hat{ε}}_{δ} > 0$ such that for any $ε \in (0, {\hat{ε}}_{δ})$ and any solution $u \in {\tilde{N}}_{ε}$ of (3.2), it holds $\begin{matrix} (5.5) & ‖ u ‖_{L^{\infty} (Λ_{ε}^{c})} < \sqrt{a} . \end{matrix}$ We argue by contradiction and assume that for some subsequence $ε_{n} \to 0$ we can obtain $u_{n} = u_{ε_{n}} \in {\tilde{N}}_{ε_{n}}$ such that $J_{ε_{n}}^{'} (u_{n}) = 0$ and $\begin{matrix} (5.6) & ‖ u_{n} ‖_{L^{\infty} (Λ_{ε}^{c})} ⩾ \sqrt{a} . \end{matrix}$ Since $J_{ε_{n}} (u_{n}) ⩽ d_{0} + h (ε_{n})$ , we can argue as in the first part of Lemma 3.4 in [6] to deduce that $J_{ε_{n}} (u_{n}) \to d_{0}$ . In view of Lemma 4.6, there exists $({\tilde{y}}_{n}) \subset R^{N}$ such that $ε_{n} {\tilde{y}}_{n} \to y_{0}$ for some $y_{0} \in M$ and $v_{n} = | u_{n} | (\cdot + {\tilde{y}}_{n})$ strongly converges in $H^{1} (R^{N}, R)$ .

Take $r > 0$ such that, for some subsequence still denoted by itself, it holds $B_{r} (ε_{n} {\tilde{y}}_{n}) \subset Λ$ for all $n \in N$ . Hence $B_{\frac{r}{ε_{n}}} ({\tilde{y}}_{n}) \subset Λ_{ε_{n}}$ $n \in N$ , and consequently $Λ_{ε_{n}}^{c} \subset B_{\frac{r}{ε_{n}}}^{c} ({\tilde{y}}_{n})$ for all $n \in N$ . By (5.2), we can find $R > 0$ such that $\begin{matrix} v_{n} (x) < \sqrt{a} for all | x | ⩾ R, n \in N, \end{matrix}$ from which we deduce that $| u_{n} (x) | < \sqrt{a}$ for any $x \in B_{R}^{c} ({\tilde{y}}_{n})$ and $n \in N$ . On the other hand, there exists $ν \in N$ such that for any $n ⩾ ν$ it holds $\begin{matrix} Λ_{ε_{n}}^{c} \subset B_{\frac{r}{ε_{n}}}^{c} ({\tilde{y}}_{n}) \subset B_{R}^{c} ({\tilde{y}}_{n}) . \end{matrix}$ Therefore, $| u_{n} (x) | < \sqrt{a}$ for any $x \in Λ_{ε_{n}}^{c}$ and $n ⩾ ν$ , and this contradicts (5.6).

Let ${\tilde{ε}}_{δ} > 0$ be given by Theorem 4.2 and we set $ε_{δ} = min {{\tilde{ε}}_{δ}, {\hat{ε}}_{δ}}$ . Applying Theorem 4.2 we obtain at least ${cat}_{M_{δ}} (M)$ nontrivial solutions to (3.2). If $u \in H_{ε}$ is one of these solutions, then $u \in {\tilde{N}}_{ε}$ , and in view of (5.5) and the definition of g, we infer that u is also a solution to (3.1). Then, ${\hat{u}}_{ε} (x) = u_{ε} (x / ε)$ is a solution to (1.1), and we deduce that (1.1) has at least ${cat}_{M_{δ}} (M)$ nontrivial solutions. We now consider $ε_{n} \to 0$ and take a sequence $(u_{n})$ of solutions to (3.2) as above. In order to investigate the behavior of the maximum points of $| u_{n} |$ , we first notice that, by $(g_{1})$ , there exists $γ \in (0, \sqrt{a})$ such that $\begin{array}{l} (5.7) & g_{ε} (x, t^{2}) t^{2} ⩽ \frac{V_{0}}{2} t^{2} for all x \in R^{N}, | t | ⩽ γ . \end{array}$ Arguing as above, we can find $R > 0$ such that $\begin{array}{l} (5.8) & ‖ u_{n} ‖_{L^{\infty} (B_{R}^{c} ({\tilde{y}}_{n}))} < γ . \end{array}$ Up to a subsequence, we may also assume that $\begin{array}{l} (5.9) & ‖ u_{n} ‖_{L^{\infty} (B_{R} ({\tilde{y}}_{n}))} ⩾ γ . \end{array}$ Indeed, if (5.9) does not hold, we get $‖ u_{n} ‖_{L^{\infty} (R^{N})} < γ$ , and using $J_{ε_{n}}^{'} (u_{n}) = 0$ , (5.7) and (2.1), we deduce that $\begin{array}{l} \int_{R^{N}} {| \nabla | u_{n} | |}^{2} + V_{0} | u_{n} |^{2} d x & ⩽ ‖ u_{n} ‖_{ε_{n}}^{2} = \int_{R^{N}} g_{ε_{n}} (x, | u_{n} |^{2}) | u_{n} |^{2} d x ⩽ \frac{V_{0}}{2} \int_{R^{N}} | u_{n} |^{2} d x . \end{array}$ This fact yields $‖ | u_{n} | ‖_{V_{0}} = 0$ , which is impossible. Hence, (5.9) is satisfied.

In the light of (5.8) and (5.9), we can see that if $p_{n}$ is a global maximum point of $| u_{n} |$ , then $p_{n}$ belongs to $B_{R} ({\tilde{y}}_{n})$ , that is $p_{n} = {\tilde{y}}_{n} + q_{n}$ for some $q_{n} \in B_{R}$ . Since the associated solution of (1.1) is of the form ${\hat{u}}_{n} (x) = u_{n} (x / ε_{n})$ , we infer that $η_{n} = ε_{n} {\tilde{y}}_{n} + ε_{n} q_{n}$ is a global maximum point of $| {\hat{u}}_{n} |$ . Using $(q_{n}) \subset B_{R}$ , $ε_{n} {\tilde{y}}_{n} \to y_{0}$ , $V (y_{0}) = V_{0}$ , and the continuity of V, we deduce that $\begin{matrix} lim_{n \to \infty} V (η_{n}) = V_{0} . \end{matrix}$ Finally, we provide the exponential decay estimate for $| {\hat{u}}_{n} |$ . Using (5.2) and $(g_{1})$ , there exists $R > 0$ such that $\begin{matrix} (5.10) & h_{n} = g (ε_{n} x + ε_{n} {\tilde{y}}_{n}, v_{n}^{2}) v_{n} ⩽ \frac{V_{0}}{2} v_{n} for | x | ⩾ R . \end{matrix}$ Moreover, by (5.4) and $(V_{1})$ , we can see that $v_{n}$ satisfies $\begin{array}{l} - Δ v_{n} + \frac{V_{0}}{2} v_{n} = h_{n} - ({\tilde{V}}_{n} - \frac{V_{0}}{2}) v_{n} ⩽ \frac{V_{0}}{2} v_{n} ⩽ 0 for | x | ⩾ R . \end{array}$ Let α and M be positive constants such that $α^{2} < \frac{V_{0}}{2}$ and $v_{n} (x) ⩽ M e^{- α R}$ for all $| x | = R$ . Then the function $w (x) = M e^{- α | x |}$ satisfies $\begin{array}{l} - Δ w + \frac{V_{0}}{2} w ⩾ (\frac{V_{0}}{2} - α^{2}) w > 0 for all x \neq 0 . \end{array}$ Set $z_{n} = w - v_{n}$ and note that it satisfies $\begin{matrix} \{\begin{matrix} - Δ z_{n} + \frac{V_{0}}{2} z_{n} ⩾ 0 & in | x | ⩾ R, \\ z_{n} ⩾ 0 & on | x | = R, \\ {lim}_{| x | \to \infty} z_{n} (x) = 0 . \end{matrix} \end{matrix}$ By maximum principle [29], we have that $z_{n} (x) ⩾ 0$ for $| x | ⩾ R$ and we deduce that $v_{n} (x) ⩽ M e^{- α | x |}$ for all $| x | ⩾ R$ and $n \in N$ . This fact combined with (5.1) implies that $v_{n} (x) ⩽ C^{'} e^{- α | x |}$ for all $x \in R^{N}$ and $n \in N$ . Recalling the definition of ${\hat{u}}_{n}$ , we obtain that $\begin{array}{l} | {\hat{u}}_{n} | (x) = | u_{n} | (\frac{x}{ε_{n}}) = v_{n} (\frac{x}{ε_{n}} - {\tilde{y}}_{n}) = v_{n} (\frac{x + ε_{n} q_{n} - η_{n}}{ε_{n}}) ⩽ C_{1} e^{- \frac{C_{2}}{ε_{n}} | x - η_{n} |} for all x \in R^{N} . \end{array}$ This ends the proof of Theorem 1.1. □

6. Proof of Theorem 1.2

In this section we deal with the following supercritical magnetic equation $\begin{matrix} (6.1) & {(\frac{1}{ı} \nabla - A_{ε} (x))}^{2} u + V_{ε} (x) u = | u |^{q - 2} u + λ | u |^{r - 2} u in R^{N}, \end{matrix}$ where $2 < q < 2^{*} < r$ . Inspired by [26], we truncate the nonlinearity $f (| u |^{2}) u = | u |^{q - 2} u + λ | u |^{r - 2} u$ as follows. Let $K > 0$ be a real number whose value will be fixed later, and we set $\begin{matrix} f_{λ} (t) = \{\begin{matrix} 0 & if t ⩽ 0, \\ t^{\frac{q - 2}{2}} + λ t^{\frac{r - 2}{2}} & if 0 < t < K, \\ (1 + λ K^{\frac{r - q}{2}}) t^{\frac{q - 2}{2}} & if t ⩾ K . \end{matrix} \end{matrix}$ It is easy to check that $f_{λ}$ satisfies assumptions $(f_{1})$ – $(f_{4})$ ( $(f_{2})$ holds with $γ = 0$ and $(f_{3})$ holds with $θ = q > 2$ ). In particular, $\begin{matrix} (6.2) & f_{λ} (t) ⩽ (1 + λ K^{\frac{r - q}{2}}) t^{\frac{q - 2}{2}} for all t ⩾ 0 . \end{matrix}$ Now we consider the following truncated problem $\begin{matrix} (6.3) & {(\frac{1}{ı} \nabla - A_{ε} (x))}^{2} u + V_{ε} (x) u = f_{λ} (| u |^{2}) u in R^{N}, \end{matrix}$ and the corresponding functional $J_{ε, λ} : H_{ε} \to R$ defined as $\begin{matrix} J_{ε, λ} (u) = \frac{1}{2} ‖ u ‖_{ε}^{2} - \frac{1}{2} \int_{R^{N}} F_{λ} (| u |^{2}) d x . \end{matrix}$ We also introduce the autonomous functional $I_{0, λ} : H_{0} \to R$ given by $\begin{matrix} I_{0, λ} (u) = \frac{1}{2} ‖ u ‖_{0}^{2} - \frac{1}{2} \int_{R^{N}} F_{λ} (u^{2}) d x . \end{matrix}$ Using Theorem 1.1, we know that for any $λ ⩾ 0$ and $δ > 0$ , there exists $\bar{ε} (λ, δ) > 0$ such that, for any $ε \in (0, \bar{ε} (λ, δ))$ , problem (6.3) admits at least ${cat}_{M_{δ}} (M)$ nontrivial solutions. Let $u_{ε, λ}$ one of these solutions. Note that, by the proof of Theorem 1.1, $u_{ε, λ}$ satisfies the following energy estimate: $\begin{matrix} J_{ε, λ} (u_{ε, λ}) ⩽ d_{0, λ} + h_{λ} (ε), \end{matrix}$ where $d_{0, λ}$ is the mountain pass level related to the functional $I_{0, λ}$ and $h_{λ} (ε) \to 0$ as $ε \to 0$ . Then, decreasing $\bar{ε} (λ, δ)$ if necessary, we may assume that $\begin{matrix} J_{ε, λ} (u_{ε, λ}) ⩽ d_{0, 0} + 1 \end{matrix}$ for any $ε \in (0, \bar{ε} (λ, δ))$ (note that $d_{0, λ} ⩽ d_{0, 0}$ for any $λ ⩾ 0$ ). Hence, using $(f_{3})$ , we can see that $\begin{array}{l} J_{ε, λ} (u_{ε, λ}) & = J_{ε, λ} (u_{ε, λ}) - \frac{1}{θ} ⟨ J_{ε, λ}^{'} (u_{ε, λ}), u_{ε, λ} ⟩ \\ = (\frac{1}{2} - \frac{1}{θ}) ‖ u_{ε, λ} ‖_{ε}^{2} + \frac{1}{θ} \int_{R^{N}} f_{λ} (| u_{ε, λ} |^{2}) | u_{ε, λ} |^{2} - \frac{θ}{2} F_{λ} (| u_{ε, λ} |^{2}) d x \\ ⩾ (\frac{1}{2} - \frac{1}{θ}) ‖ u_{ε, λ} ‖_{ε}^{2} . \end{array}$ Consequently, $\begin{array}{l} (6.4) & ‖ u_{ε, λ} ‖_{ε} ⩽ {[(\frac{2 θ}{θ - 2}) (d_{0, 0} + 1)]}^{\frac{1}{2}} = \bar{C} for all ε \in (0, \bar{ε} (λ, δ)) . \end{array}$ At this point, we claim that $u_{ε, λ}$ is a solution of the original problem (6.1) as long as $λ > 0$ is sufficiently small. More precisely, we need to show that we can find $K_{0} > 0$ such that for any $K ⩾ K_{0}$ , there exists $λ_{0} = λ_{0} (K) > 0$ such that $\begin{matrix} (6.5) & ‖ u_{ε, λ} ‖_{L^{\infty} (R^{N})} ⩽ K for all λ \in [0, λ_{0}] . \end{matrix}$ We shall assume that $\bar{ε} (λ, δ)$ is small in such a way that (6.4) holds true. In what follows, we write u instead of $u_{ε, λ}$ . For any $L > 0$ , we define $u_{L} = min {| u |, L}$ and $w_{L} = | u | u_{L}^{β - 1}$ , where $β > 1$ will be chosen later. Taking $z_{L} = u_{L}^{2 (β - 1)} u$ in (6.3) we can see that $\begin{array}{l} (6.6) & ℜ (\int_{R^{N}} \nabla_{ε} u \cdot \overline{\nabla_{ε} z_{L}} d x) + \int_{R^{N}} V_{ε} (x) | u |^{2} u_{L}^{2 (β - 1)} d x = \int_{R^{N}} f_{λ} (| u |^{2}) | u |^{2} u_{L}^{2 (β - 1)} d x . \end{array}$ A direct calculation and (2.1) show that $\begin{array}{l} ℜ (\nabla_{ε} u \cdot \overline{\nabla_{ε} z_{L}}) ⩾ u_{L}^{2 (β - 1)} | \nabla_{ε} u |^{2} ⩾ u_{L}^{2 (β - 1)} {| \nabla | u | |}^{2}, \end{array}$ which combined with (6.2), (6.6) and $(V_{1})$ yields $\begin{array}{l} (6.7) & \int_{R^{N}} u_{L}^{2 (β - 1)} {| \nabla | u | |}^{2} d x ⩽ C_{λ, K} \int_{R^{N}} | u |^{q} u_{L}^{2 (β - 1)} d x, \end{array}$ where $C_{λ, K} = 1 + λ K^{\frac{r - q}{2}}$ . On the other hand, we note that $\begin{array}{l} \int_{R^{N}} | \nabla w_{L} |^{2} d x & = \int_{{| u | ⩽ L}} | \nabla w_{L} |^{2} d x + \int_{{| u | > L}} | \nabla w_{L} |^{2} d x \\ = \int_{{| u | ⩽ L}} {| \nabla | u |^{β} |}^{2} d x + \int_{{| u | > L}} L^{2 (β - 1)} {| \nabla | u | |}^{2} d x \\ ⩽ β^{2} \int_{R^{N}} u_{L}^{2 (β - 1)} {| \nabla | u | |}^{2} d x . \end{array}$ Then, using the Sobolev embedding $D^{1, 2} (R^{N}, R) \subset L^{2^{*}} (R^{N}, R)$ and (6.7), we get $\begin{array}{l} (6.8) & \begin{aligned} ‖ w_{L} ‖_{L^{2^{*}} (R^{N})}^{2} & ⩽ S_{*}^{- 1} \int_{R^{N}} | \nabla w_{L} |^{2} d x \\ ⩽ S_{*}^{- 1} β^{2} \int_{R^{N}} u_{L}^{2 (β - 1)} {| \nabla | u | |}^{2} d x \\ ⩽ S_{*}^{- 1} β^{2} C_{λ, K} \int_{R^{N}} | u |^{q} u_{L}^{2 (β - 1)} d x . \end{aligned} \end{array}$ Observing that $| u |^{q} u_{L}^{2 (β - 1)} = | u |^{q - 2} w_{L}^{2}$ , it follows from (6.8) and the Hölder inequality that $\begin{array}{l} (6.9) & ‖ w_{L} ‖_{L^{2^{*}} (R^{N})}^{2} ⩽ C_{1} β^{2} C_{λ, K} {(\int_{R^{N}} | u |^{2^{*}} d x)}^{\frac{q - 2}{2^{*}}} {(\int_{R^{N}} w_{L}^{α_{*}} d x)}^{\frac{2}{α_{*}}} \end{array}$ where $\begin{matrix} α^{*} = \frac{22^{*}}{2^{*} - (q - 2)} \in (2, 2^{*}), \end{matrix}$ and $C_{1} = S_{*}^{- 1}$ . Then, using $H^{1} (R^{N}, R) \subset L^{2^{*}} (R^{N}, R)$ , (2.1) and (6.4), we obtain $\begin{array}{l} (6.10) & ‖ w_{L} ‖_{L^{2^{*}} (R^{N})}^{2} ⩽ C_{2} β^{2} C_{λ, K} {\bar{C}}^{q - 2} ‖ w_{L} ‖_{L^{α^{*}} (R^{N})}^{2}, \end{array}$ where $C_{2} = C_{1} S_{*}^{- \frac{q - 2}{2}}$ . Let us note that, if $| u |^{β} \in L^{α^{*}} (R^{N}, R)$ , the definition of $w_{L}$ , $u_{L} ⩽ | u |$ and (6.10) imply that $\begin{array}{l} (6.11) & ‖ w_{L} ‖_{L^{2^{*}} (R^{N})}^{2} ⩽ C_{2} β^{2} C_{λ, K} {\bar{C}}^{q - 2} {(\int_{R^{N}} | u |^{β α^{*}} d x)}^{\frac{2}{α^{*}}} < \infty . \end{array}$ Taking the limit as $L \to \infty$ in (6.11) and using Fatou’s lemma, we have $\begin{array}{l} (6.12) & ‖ u ‖_{L^{β 2^{*}} (R^{N})} ⩽ {(C_{3} C_{λ, K})}^{\frac{1}{2 β}} β^{\frac{1}{β}} ‖ u ‖_{L^{β α^{*}} (R^{N})} \end{array}$ provided that $| u |^{β α^{*}} \in L^{1} (R^{N}, R)$ , where $C_{3} = C_{2} {\bar{C}}^{q - 2}$ .

Set $β = \frac{2^{*}}{α^{*}} > 1$ and we note that, since $| u | \in L^{2^{*}} (R^{N}, R)$ , the above inequality holds for this choice of β. Then, observing that $β^{2} α^{*} = β 2^{*}$ , it follows that (6.12) holds with β replaced by $β^{2}$ , so we have $\begin{array}{l} ‖ u ‖_{L^{β^{2} 2^{*}} (R^{N})} ⩽ {(C_{3} C_{λ, K})}^{\frac{1}{2 β^{2}}} β^{\frac{2}{β^{2}}} ‖ u ‖_{L^{β^{2} α^{*}} (R^{N})} ⩽ {(C_{3} C_{λ, K})}^{\frac{1}{2} (\frac{1}{β} + \frac{1}{β^{2}})} β^{\frac{1}{β} + \frac{2}{β^{2}}} ‖ u ‖_{L^{β α^{*}} (R^{N})} . \end{array}$ Iterating this process and using the fact that $β α^{*} = 2^{*}$ , we deduce that for every $m \in N$ $\begin{array}{l} (6.13) & ‖ u ‖_{L^{β^{m} 2^{*}} (R^{N})} ⩽ {(C_{3} C_{λ, K})}^{\sum_{j = 1}^{m} \frac{1}{2 β^{j}}} β^{\sum_{j = 1}^{m} j β^{- j}} ‖ u ‖_{L^{2^{*}} (R^{N})} . \end{array}$ Taking the limit as $m \to \infty$ in (6.13) and using the embedding $D^{1, 2} (R^{N}, R) \subset L^{2^{*}} (R^{N}, R)$ , (2.1) and (6.4), we obtain $\begin{array}{l} (6.14) & ‖ u ‖_{L^{\infty} (R^{N})} ⩽ {(C_{3} C_{λ, K})}^{γ_{1}} β^{γ_{2}} C_{4}, \end{array}$ where $C_{4} = S_{*}^{- \frac{1}{2}} \bar{C}$ , and $\begin{matrix} γ_{1} = \frac{1}{2} \sum_{j = 1}^{\infty} \frac{1}{β^{j}} < \infty and γ_{2} = \sum_{j = 1}^{\infty} \frac{j}{β^{j}} < \infty . \end{matrix}$ Next, we will find suitable values of K and λ such that the following inequality holds $\begin{matrix} {(C_{3} C_{λ, K})}^{γ_{1}} β^{γ_{2}} C_{4} ⩽ K, \end{matrix}$ or equivalently $\begin{matrix} 1 + λ K^{\frac{r - q}{2}} ⩽ C_{3}^{- 1} β^{- \frac{γ_{2}}{γ_{1}}} {(K C_{4}^{- 1})}^{\frac{1}{γ_{1}}} . \end{matrix}$ Take $K > 0$ such that $\begin{matrix} \frac{{(K C_{4}^{- 1})}^{\frac{1}{γ_{1}}}}{C_{3} β^{\frac{γ_{2}}{γ_{1}}}} - 1 > 0, \end{matrix}$ and fix $λ_{0} > 0$ such that $\begin{matrix} 0 ⩽ λ ⩽ λ_{0} ⩽ [\frac{{(K C_{4}^{- 1})}^{\frac{1}{γ_{1}}}}{C_{3} β^{\frac{γ_{2}}{γ_{1}}}} - 1] \frac{1}{K^{\frac{r - q}{2}}} . \end{matrix}$ Then, using (6.14), we can conclude that $\begin{matrix} ‖ u ‖_{L^{\infty} (R^{N})} ⩽ K for all λ \in [0, λ_{0}] . \end{matrix}$ This completes the proof of Theorem 1.2.

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