Existence and asymptotic behavior of solutions for a class of semilinear subcritical elliptic systems

Abstract

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system $\begin{matrix} \{\begin{matrix} - ε^{2} Δ u + W (x) u = Q_{u} (u, v) in R^{N}, \\ - ε^{2} Δ v + V (x) v = Q_{v} (u, v) in R^{N}, \\ u, v \in H^{1} (R^{N}), u (x), v (x) > 0 for each x \in R^{N}, \end{matrix} \end{matrix}$ where $ε > 0$ , W and V are positive potentials of $C^{2}$ class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme.

Keywords

Elliptic system asymptotic analysis Palais–Smale condition positive solution

1. Introduction

Consider the nonlinear Schrödinger equations (NLS in short) $\begin{matrix} (NSL) & \{\begin{matrix} - ε^{2} Δ u + V (x) u = f (u) in R^{N}, \\ u > 0 in R^{N} \end{matrix} \end{matrix}$ with f continuous. The case with f superlinear has been widely investigated in the past decades. In case $\begin{array}{rcl} (1.1) & 0 < inf_{x \in R^{N}} V (x) < \underset{| x | \to \infty}{lim inf} V (x), \end{array}$ the problem was considered by Rabinowitz in his seminal paper [22] with subsequent improvements by Del Pino and Felmer [9] that considered the $\begin{array}{rcl} 0 < V_{0} = inf_{x \in Ω} V (x) < min_{x \in \partial Ω} V (x), for some domain Ω \subset R^{N}, \end{array}$ which is more weaker than (1.1).

Since then a number of authors have considered this type of problem with another geometries on the potential V. We refer the reader to [7,20], for the case with V satisfying a 1-periodicity condition and [10] for the case where potential V has saddle geometry like. In [2], Alves has studied two new geometries on the potential V, which we will consider in this article and detail in the hypotheses $(A_{0})$ , $(A_{1})$ , $(A_{2})$ and $(A_{3})$ below.

In this paper we consider a system of coupled NLS equations, which arise for example in nonlinear optics, as can be seem in [4]. We show existence of a positive solution for the following system given by $\begin{matrix} (S_{ε}) & \{\begin{matrix} - ε^{2} Δ u + W (x) u = Q_{u} (u, v) in R^{N}, \\ - ε^{2} Δ v + V (x) v = Q_{v} (u, v) in R^{N}, \\ u, v \in H^{1} (R^{N}), u (x), v (x) > 0 for each x \in R^{N}, \end{matrix} \end{matrix}$ where $ε > 0$ , $N ⩾ 3$ and W, V are potentials of $C^{2}$ class.

More specifically, the potentials W and V satisfying two classes of potentials, namely:

Class 1: Potentials W and V verify the Palais–Smale condition.

There are $W_{0}, V_{0} > 0$ such that $W (x) ⩾ W_{0}$ and $V (x) ⩾ V_{0}$ , $\forall x \in R^{N}$ .

$W, V \in C^{2} (R^{N})$ and W, V, $\frac{\partial W}{\partial x_{i}}$ , $\frac{\partial V}{\partial x_{i}}$ , $\frac{\partial^{2} W}{\partial x_{i} \partial x_{j}}$ and $\frac{\partial^{2} V}{\partial x_{i} \partial x_{j}}$ are bounded in $R^{N}$ for all $i, j \in {1, \dots, N}$ .

The potentials W and V verify the Palais–Smale condition, that is, if $(x_{n}) \subset R^{N}$ is such that $(W (x_{n}))$ and $(V (x_{n}))$ are bounded and $\nabla W (x_{n}) \to 0$ and $\nabla V (x_{n}) \to 0$ , then $(x_{n})$ possesses a convergent subsequence in $R^{N}$ .

Class 2: Potentials W and V do not have critical point on the boundary of some bounded domain.

In this class of potential, we suppose that W, V verify $(A_{0})$ – $(A_{1})$ and the following additional condition:

There is a bounded domain $Λ \subset R^{N}$ such that $\nabla W (x) \neq 0$ and $\nabla V (x) \neq 0$ , for all $x \in \partial Λ$ .

Setting $R_{+}^{2} : = [0, \infty] \times [0, \infty]$ , the function $Q \in C^{2} (R_{+}^{2}, R)$ is a homogeneous function of degree p. More precisely, the hypotheses on the nonlinearity Q are the following:

There is $2 < p < 2^{*} : = 2 N / (N - 2)$ such that $\begin{matrix} Q (t u, t v) = t^{p} Q (u, v) for each t > 0, (u, v) \in R_{+}^{2} . \end{matrix}$

There is $c_{1} > 0$ such that $\begin{matrix} | Q_{u} (u, v) | + | Q_{v} (u, v) | ⩽ c_{1} (u^{p - 1} + v^{p - 1}) for each (u, v) \in R_{+}^{2} . \end{matrix}$

$Q_{u} (0, 1) = 0$ , $Q_{v} (1, 0) = 0$ .

$Q_{u} (1, 0) = 0$ , $Q_{v} (0, 1) = 0$ .

$Q (u, v) > 0$ for each $u, v > 0$ .

$Q_{u} (u, v), Q_{v} (u, v) ⩾ 0$ for each $(u, v) \in R_{+}^{2}$ .

For each $ε > 0$ , a pair $(u_{ε}, v_{ε}) \in H^{1} (R^{N}) \times H^{1} (R^{N})$ is a positive solution of system $(S_{ε})$ if $u_{ε} > 0$ and $v_{ε} > 0$ a.e in $R^{N}$ and $\begin{array}{l} ε^{2} \int_{R^{N}} \nabla u_{ε} \nabla ϕ d x + ε^{2} \int_{R^{N}} \nabla v_{ε} \nabla ψ d x + \int_{R^{N}} W (x) u_{ε} ϕ d x + \int_{R^{N}} V (x) v_{ε} ψ d x \\ = \int_{R^{N}} [ϕ Q_{u} (u_{ε}, v_{ε}) + ψ Q_{v} (u_{ε}, v_{ε})] d x, \end{array}$ for all $(ϕ, ψ) \in H^{1} (R^{N}) \times H^{1} (R^{N})$ .

In our main result we use a suitable truncation to apply a Del Pino and Felmer’s approach [9] combined with the Mountain Pass Theorem.

Theorem 1.1.
Suppose that W and V belong to Classe 1 or 2 and Q satisfies $(Q_{0})$ – $(Q_{5})$ . Then, system $(S_{ε})$ has a positive solution for $ε > 0$ small enough.

Concerning the class of nonlinearities we are dealing, we have the following examples from [8]. Let $q ⩾ 1$ and $\begin{matrix} P_{q} (s, t) = \sum_{α_{i} + β_{i} = q} a_{i} s^{α_{i}} t^{β_{i}}, \end{matrix}$ where $i \in {1, \dots, k}$ , $α_{i}, β_{i} ⩾ 1$ and $a_{i} \in R$ . The following functions and its possible combinations, with appropriated choices of the coefficients $a_{i}$ , satisfy our hypothesis on Q $\begin{matrix} Q_{1} (s, t) = P_{p} (s, t), Q_{2} (s, t) = \sqrt[r]{P_{l} (s, t)} and Q_{3} (s, t) = \frac{P_{l_{1}} (s, t)}{P_{l_{2}} (s, t)}, \end{matrix}$ with $r = p l$ and $l_{1} - l_{2} = p$ .

We observe that there is an extensive bibliography on the study of elliptic systems (see [6,11,12,24] and references therein). We refer [14,23,25] for results in the spirit of the paper [22] is derived and [1,3,15,16,18] for results in the spirit of the paper [9] is obtained.

In [13] the authors consider a coupled elliptic system of equations involving fractional Laplace operator. The qualitative behavior of solutions of such system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. In [13] the authors derive monotonicity formulae for entire solutions of the system. They apply these formulae to give a classification of finite Morse index solutions. In the end, they provide an open problem in regards to monotonicity formulae for Lane–Emden systems. For more information of Variational methods and Morse index for semilinear problems or for gradient system, see Chapters 5 and 6 in [21].

The present work is strongly influenced by the articles above. Below we list what we believe that are the main contributions of our paper.

We give a result of existence of solutions for a system of coupled NLS equations considering the two new geometries on the potentials, much in the spirit of Alves’s [2] one for the single equation case.

A crucial step of the proof will consist to obtain some estimates from above for the $H^{1} (R^{N}) \times H^{1} (R^{N})$ -norm and $L^{\infty} (R^{N}) \times L^{\infty} (R^{N})$ -norm of the family of solutions of an auxiliary system obtained using arguments that can be found in [1] (see Secion 4).

The paper is organized as follows. In order to overcome the lack of compactness, in Section 2 we make a penalization of the nonlinearity using arguments that can be found in [1]. In Section 3 we show existence of solution for the auxiliary system introduced in Section 2. In Section 4 we obtain uniform estimates in order to show that the solution of the auxiliary system is a solution of the original system. The proof of the main result is in Section 5.
2. Variational framework and a modified system

Since we are interested in positive solutions we extend the function Q to the whole $R^{2}$ by setting $Q (u, v) = 0$ if $u ⩽ 0$ or $v ⩽ 0$ . We also note that, since Q is p-homogeneous, for each $(s, t) \in R^{2}$ we have $\begin{matrix} p Q (s, t) = s Q_{s} (s, t) + t Q_{t} (s, t) \end{matrix}$ and $\begin{matrix} p (p - 1) Q (s, t) = s^{2} Q_{s s} (s, t) + t^{2} Q_{t t} (s, t) + 2 s t Q_{s t} (s, t) . \end{matrix}$

Using the change variable $v (x) = u (ε x)$ , it is possible to prove that $(S_{ε})$ is equivalent to the following system $\begin{matrix} ({\hat{S}}_{ε}) & \{\begin{matrix} - Δ u + W (ε x) u = Q_{u} (u, v) in R^{N}, \\ - Δ v + V (ε x) v = Q_{v} (u, v) in R^{N}, \\ u, v \in H^{1} (R^{N}), u (x), v (x) > 0 for each x \in R^{N} . \end{matrix} \end{matrix}$

In this section, we use a penalization method found in [9] to overcome the lack of compactness originated by the unboundedness of $R^{N}$ . Here, we fix some notations and we study an adaptation for a special class of elliptic system introduced by [1].

We start by choosing $α > 0$ and considering $η : R \to R$ a non-increasing function of class $C^{2}$ such that $\begin{matrix} η \equiv 1 on (- \infty, α], η \equiv 0 on [5 α, + \infty), | η^{'} (s) | ⩽ \frac{C}{α} and | η^{″} (s) | ⩽ \frac{C}{α^{2}}, \end{matrix}$ for each $s \in R$ and for some positive constant $C > 0$ independent on α. Using the function η, we define $\hat{Q} : R^{2} \to R$ by $\begin{matrix} \hat{Q} (s, t) : = η (| (s, t) |) Q (s, t) + (1 - η (| (s, t) |)) A (s^{2} + t^{2}), \end{matrix}$ where $\begin{matrix} A : = max {\frac{Q (s, t)}{s^{2} + t^{2}} : (s, t) \in R^{2}, α ⩽ | (s, t) | ⩽ 5 α} . \end{matrix}$ Notice that, since $A > 0$ tends to zero as $α \to 0^{+}$ , we may suppose that $A < min {W_{0}, V_{0}}$ .

Finally, given a bounded domain $Ω \subset R^{N}$ , denoting by $I_{Ω}$ the characteristic function of the set Ω, we define $H : R^{N} \times R^{2} \to R$ by setting $\begin{array}{rcl} (2.1) & H (x, s, t) : = I_{Ω} (x) Q (s, t) + (1 - I_{Ω} (x)) \hat{Q} (s, t) . \end{array}$

For future reference we note that arguing as in [1, Lemma 2.2], for any $α > 0$ small and $(s, t) \in R^{2}$ we have the following result.

Lemma 2.1.
The function H satisfies the following estimates:
$p H (x, s, t) = s H_{s} (x, s, t) + t H_{t} (x, s, t)$ , for each $x \in Ω$ .

$2 H (x, s, t) ⩽ s H_{s} (x, s, t) + t H_{t} (x, s, t)$ , for each $x \in R^{N} ∖ Ω$ .

For α small, we have $\begin{matrix} s H_{s} (x, s, t) + t H_{t} (x, s, t) ⩽ \frac{1}{4} (W_{0} s^{2} + V_{0} t^{2}), for each x \in R^{N} ∖ Ω . \end{matrix}$

For α small, we have $\begin{matrix} \frac{| H_{s} (x, s, t) |}{α}, \frac{| H_{t} (x, s, t) |}{α} ⩽ \frac{min {W_{0}, V_{0}}}{4}, for each x \in R^{N} ∖ Ω . \end{matrix}$

From now on we assume that α is chosen in such way that the last above inequality holds.

In view by definition (2.1), we deal in the sequel with the modified system $\begin{matrix} (S_{ε, aux}) & \{\begin{matrix} - Δ u + W (ε x) u = H_{u} (ε x, u, v) in R^{N}, \\ - Δ v + V (ε x) v = H_{v} (ε x, u, v) in R^{N}, \\ u, v \in H^{1} (R^{N}) \end{matrix} \end{matrix}$ and we will look for solutions $(u_{ε}, v_{ε}) \in H^{1} (R^{N}) \times H^{1} (R^{N})$ verifying $\begin{matrix} | (u_{ε} (ε x), v_{ε} (ε x)) | ⩽ α for each x \in R^{N} ∖ Ω_{ε}, \end{matrix}$ where $Ω_{ε} : = {x \in R^{N} : ε x \in Ω}$ .

It follows from $(A_{0})$ and $(Q_{1})$ that the critical points of the $C^{1}$ -functional $J_{ε} : H^{1} (R^{N}) \times H^{1} (R^{N}) \to R$ given by $\begin{matrix} J_{ε} (u, v) : = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + | \nabla v |^{2} + W (ε x) | u |^{2} + V (ε x) | v |^{2}) d x - \int_{R^{N}} H (ε x, u, v) d x \end{matrix}$ are weak solutions of $(S_{ε, aux})$ . Note that from $(A_{0})$ and $(A_{1})$ we can work in $H^{1} (R^{N}) \times H^{1} (R^{N})$ with the norm $\begin{matrix} {‖ (u, v) ‖}^{2} : = \int_{R^{N}} (| \nabla u |^{2} + | \nabla v |^{2} + W (ε x) | u |^{2} + V (ε x) | v |^{2}) d x, \end{matrix}$ which is equivalent to usual norm.

In the sequel, let us denote by $I_{\infty} : H^{1} (R^{N}) \times H^{1} (R^{N}) \to R$ the functional $\begin{matrix} I_{\infty} (u, v) = \frac{1}{2} \int_{R^{N}} (| \nabla u |^{2} + | \nabla v |^{2} + W_{\infty} | u |^{2} + V_{\infty} | v |^{2}) d x - \int_{R^{N}} Q (u, v) d x, \end{matrix}$ where $\begin{matrix} W_{\infty} = max_{x \in R^{N}} W (x) \end{matrix}$ and $\begin{matrix} V_{\infty} = max_{x \in R^{N}} V (x) . \end{matrix}$

Furthermore, let us denote by $c_{\infty}$ the mountain level associated with $I_{\infty}$ , that is, $\begin{matrix} c_{\infty} = inf_{γ \in Γ} max_{t \in [0, 1]} I_{\infty} (γ (t)), \end{matrix}$ where $\begin{matrix} Γ = {γ \in C ([0, 1], H^{1} (R^{N}) \times H^{1} (R^{N})) : γ (0) = (0, 0) and I_{\infty} (γ (1)) < 0} . \end{matrix}$

Here, we would like point out that $c_{\infty}$ depends only on $W_{\infty}$ , $V_{\infty}$ and Q.
3. Existence of a ground state solution for system $(S_{ε, aux})$

The main result in this section is:

Theorem 3.1.
Suppose that W and V satisfy $(A_{0})$ – $(A_{1})$ . Suppose also that Q satisfies $(Q_{0})$ – $(Q_{5})$ . Then, for all $ε > 0$ , system $(S_{ε, aux})$ has a ground state positive solution.

In order to show existence of a ground state solution for the modified system $(S_{ε, aux})$ , we use the Mountain Pass Theorem [5].
Lemma 3.2.
The functional $J_{ε}$ satisfies the following conditions
There are $C, ρ > 0$ , such that $\begin{matrix} J_{ε} (u, v) ⩾ C, if ‖ (u, v) ‖ = ρ . \end{matrix}$

For any $(ϕ, ψ) \in C_{0}^{\infty} (Ω_{ε}) \times C_{0}^{\infty} (Ω_{ε})$ with $ϕ, ψ ⩾ 0$ , we have $\begin{matrix} lim_{t \to \infty} J_{ε} (t ϕ, t ψ) = - \infty . \end{matrix}$

Proof.
Using $(H_{1})$ , $(Q_{1})$ , $(H_{2})$ and $(H_{3})$ , we have $\begin{matrix} J_{ε} (u, v) ⩾ \frac{1}{2} {‖ (u, v) ‖}^{2} - \frac{c_{1}}{p} \int_{Ω_{ε}} (| u |^{p} + | v |^{p}) d x - \frac{1}{8} \int_{R^{N} ∖ Ω_{ε}} (W_{0} | u |^{2} + V_{0} | v |^{2}) d x . \end{matrix}$

By Sobolev embeddings, there is $C > 0$ such that $\begin{matrix} J_{ε} (u, v) ⩾ \frac{3}{8} {‖ (u, v) ‖}^{2} - \frac{C}{p} {‖ (u, v) ‖}^{p} \end{matrix}$ and the proof of item $(i)$ is over. Now, by definition of H and $(Q_{0})$ , we get $\begin{matrix} J_{ε} (t ϕ, t ψ) = \frac{t^{2}}{2} {‖ (ϕ, ψ) ‖}^{2} - t^{p} \int_{Ω_{ε}} Q (ϕ, ψ) d x \end{matrix}$ and the proof of item $(ii)$ is over. □

Hence, there is a Palais–Smale sequence $(u_{n}, v_{n}) \subset H^{1} (R^{N}) \times H^{1} (R^{N})$ at level $c_{ε}$ , where $c_{ε}$ is the minimax level associated to $J_{ε}$ . Lemma 3.3.
The functional $J_{ε}$ satisfies the Palais–Smale condition at any level c.
Proof.
Let $(u_{n}, v_{n}) \subset H^{1} (R^{N}) \times H^{1} (R^{N})$ such that $J_{ε} (u_{n}, v_{n}) \to c$ and $J_{ε}^{'} (u_{n}, v_{n}) = o_{n} (1)$ . Then, from $(H_{1})$ we get $\begin{array}{l} c + o_{n} (1) ‖ (u_{n}, v_{n}) ‖ + o_{n} (1) \\ ⩾ J_{ε} (u_{n}, v_{n}) - \frac{1}{p} J_{ε}^{'} (u_{n}, v_{n}) (u_{n}, v_{n}) \\ = (\frac{1}{2} - \frac{1}{p}) {‖ (u_{n}, v_{n}) ‖}^{2} \\ - \int_{R^{N} ∖ Ω_{ε}} (H (ε x, u_{n}, v_{n}) - \frac{1}{p} [u_{n} H_{u} (ε x, u_{n}, v_{n}) + v_{n} H_{v} (ε x, u_{n}, v_{n})]) d x . \end{array}$

From $(H_{2})$ , we have $\begin{array}{l} (\frac{1}{2} - \frac{1}{p}) {‖ (u_{n}, v_{n}) ‖}^{2} & ⩽ c + o_{n} (1) ‖ (u_{n}, v_{n}) ‖ + o_{n} (1) \\ + (\frac{1}{2} - \frac{1}{p}) \int_{R^{N} ∖ Ω_{ε}} [u_{n} H_{u} (ε x, u_{n}, v_{n}) + v_{n} H_{v} (ε x, u_{n}, v_{n})] d x . \end{array}$

Using $(H_{3})$ , we obtain $\begin{array}{rcl} \frac{3}{4} (\frac{1}{2} - \frac{1}{p}) {‖ (u_{n}, v_{n}) ‖}^{2} ⩽ c + o_{n} (1) ‖ (u_{n}, v_{n}) ‖ + o_{n} (1), \end{array}$ which implies that $(u_{n}, v_{n})$ is bounded in $H^{1} (R^{N}) \times H^{1} (R^{N})$ . Then, up to a subsequence, we may suppose that, $\begin{array}{l} (u_{n}, v_{n}) ⇀ (u, v) weakly in H^{1} (R^{N}) \times H^{1} (R^{N}), \\ u_{n} \to u, v_{n} \to v strongly in L_{loc}^{s} (R^{N}) for any 2 ⩽ s < 2^{*}, \\ u_{n} (x) \to u (x), v_{n} (x) \to v (x) a.e for x \in R^{N} . \end{array}$

Now using a density argument, we can conclude that $(u, v)$ is a critical point of $J_{ε}$ .

From [1, Lemma 3.2 inequality (3.3)], for any $ξ >$ given, there is $R > 0$ such that $\begin{matrix} \underset{n \to \infty}{lim sup} \int_{R^{N} ∖ B_{R}} [| \nabla u_{n} |^{2} + | \nabla v_{n} |^{2} + W (ε x) | u_{n} |^{2} + V (ε x) | v_{n} |^{2}] d x < ξ, \end{matrix}$ where $B_{R}$ denotes the ball with center 0 and radius R. This inequality and $(H_{3})$ imply that, for n large enough, there holds $\begin{matrix} (3.1) & \int_{R^{N} ∖ B_{R}} [u_{n} H_{u} (ε x, u_{n}, v_{n}) + v_{n} H_{v} (ε x, u_{n}, v_{n})] d x ⩽ \frac{1}{4} ξ . \end{matrix}$

On the other hand, taking R large enough, we can suppose that $\begin{matrix} (3.2) & | \int_{R^{N} ∖ B_{R}} [u H_{u} (ε x, u, v) + v H_{v} (ε x, u, v)] d x | < ξ . \end{matrix}$ Then, by (3.1) and (3.2), we can conclude $\begin{array}{l} \int_{R^{N} ∖ B_{R}} [u_{n} H_{u} (ε x, u_{n}, v_{n}) + v_{n} H_{v} (ε x, u_{n}, v_{n})] d x \\ = \int_{R^{N} ∖ B_{R}} [u H_{u} (ε x, u, v) + v H_{v} (ε x, u, v)] d x + o_{n} (1) \end{array}$ and thus, $\begin{array}{l} \int_{R^{N}} [u_{n} H_{u} (ε x, u_{n}, v_{n}) + v_{n} H_{v} (ε x, u_{n}, v_{n})] d x \\ = \int_{R^{N}} [u H_{u} (ε x, u, v) + v H_{v} (ε x, u, v)] d x + o_{n} (1) . \end{array}$

The last equality implies $\begin{matrix} {‖ (u_{n}, v_{n}) ‖}^{2} = {‖ (u, v) ‖}^{2} + o_{n} (1) . \end{matrix}$ The proof is now complete. □

3.1. Proof of Theorem 3.1

Proof.
The proof is a consequence of Lemma 3.2, Lemma 3.3 and Mountain Pass Theorem [5]. □
4. Some uniform estimates

The lemmas below establishes an important estimates from above for the $H^{1} (R^{N}) \times H^{1} (R^{N})$ -norm and $L^{\infty} (R^{N}) \times L^{\infty} (R^{N})$ -norm of the family of solution of system $(S_{ε, aux})$ denoted by $(u_{ε}, v_{ε})$ .

Lemma 4.1.
For all $ε > 0$ , the solution $(u_{ε}, v_{ε})$ of $(S_{ε, aux})$ satisfies the estimate $\begin{matrix} {‖ (u_{ε}, v_{ε}) ‖}^{2} ⩽ \frac{8}{3} \frac{p}{p - 2} c_{\infty} . \end{matrix}$
Proof.
Using the fact that $(u_{ε}, v_{ε})$ is a critical point of $J_{ε}$ , we must have $\begin{matrix} c_{ε} = J_{ε} (u_{ε}, v_{ε}) = J_{ε} (u_{ε}, v_{ε}) - \frac{1}{p} J_{ε}^{'} (u_{ε}, v_{ε}) (u_{ε}, v_{ε}) . \end{matrix}$

Arguing as Lemma 3.3, we have $\begin{array}{rcl} \frac{3}{4} (\frac{1}{2} - \frac{1}{p}) {‖ (u_{ε}, v_{ε}) ‖}^{2} ⩽ c_{ε} . \end{array}$

Recalling that $J_{ε} (u, v) ⩽ I_{\infty} (u, v)$ , for all $(u, v) \in H^{1} (R^{N}) \times H^{1} (R^{N})$ , we conclude that $c_{ε} ⩽ c_{\infty}$ . Then, $\begin{array}{rcl} {‖ (u_{ε}, v_{ε}) ‖}^{2} ⩽ \frac{8}{3} \frac{p}{p - 2} c_{\infty} . \end{array}$ □

In the next result we establishes an important estimates from above for the $L^{\infty} (R^{N}) \times L^{\infty} (R^{N})$ -norm of the family of solution $(u_{ε}, v_{ε})$ . The key ingredient is the following result, whose proof uses an adaptation of the arguments found in [17], which are related with the Moser iteration method [19]. Lemma 4.2.
Let $(ε_{n}) \subset (0, + \infty)$ be a sequence such that $ε_{n} \to 0$ and for each $n \in N$ , let $(u_{n}, v_{n}) \subset N_{ε_{n}}$ be a solution of system $(S_{ε, aux})$ . Then, for all $(x_{n}) \subset R^{N}$ , there is $\tilde{K} > 0$ such that $\begin{matrix} | w_{n} |_{L^{\infty} (R^{N})} < \tilde{K}, for all n \in N \end{matrix}$ and $\begin{matrix} | z_{n} |_{L^{\infty} (R^{N})} < \tilde{K}, for all n \in N, \end{matrix}$ where $(w_{n} (x), z_{n} (x)) = (u_{n} (x + x_{n}), v_{n} (x + x_{n}))$ .
Proof.
For each $n \in N$ and $L > 0$ , we define $w_{L, n}, z_{L, n} \in H^{1} (R^{N}, R)$ by setting $\begin{matrix} w_{L, n} (x) : = min {w_{n} (x), L}, Υ_{w, L, n} : = w_{L, n}^{2 (β - 1)} w_{n} \end{matrix}$ and $\begin{matrix} z_{L, n} (x) : = min {z_{n} (x), L}, Υ_{z, L, n} : = z_{L, n}^{2 (β - 1)} z_{n}, \end{matrix}$ with $β > 1$ to be determined later.

Since $R^{N}$ is invariant by translation, by definition of $Υ_{w, L, n}$ and $Υ_{z, L, n}$ , we get $J_{ε_{n}}^{'} (w_{n}, z_{n}) (Υ_{w, L, n}, Υ_{z, L, n}) = 0$ . Considering that $\begin{matrix} 2 (β - 1) \int_{R^{N}} w_{L, n}^{2 (β - 1) - 1} w_{n} \nabla w_{n} \nabla w_{L, n} d x ⩾ 0 \end{matrix}$ and $\begin{matrix} 2 (β - 1) \int_{R^{N}} z_{L, n}^{2 (β - 1) - 1} z_{n} \nabla z_{n} \nabla z_{L, n} d x ⩾ 0, \end{matrix}$ we have that $\begin{matrix} (4.1) & \begin{array}{l} \int_{R^{N}} w_{L, n}^{2 (β - 1)} | \nabla w_{n} |^{2} d x + \int_{R^{N}} z_{L, n}^{2 (β - 1)} | \nabla z_{n} |^{2} d x \\ ⩽ \int_{R^{N}} (H_{w} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}) - W_{0} w_{n}) w_{n} w_{L, n}^{2 (β - 1)} d x \\ + \int_{R^{N}} (H_{z} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}) - V_{0} z_{n}) z_{n} z_{L, n}^{2 (β - 1)} d x . \end{array} \end{matrix}$

In view of $(Q_{1})$ and $(H_{3})$ we can obtain $C_{1} > 0$ such that $\begin{matrix} H_{s} (x, s, t) + H_{t} (x, s, t) ⩽ \frac{W_{0}}{4} s + \frac{V_{0}}{4} t + C_{1} (| s |^{p - 1} + | t |^{p - 1}), \end{matrix}$ for any $(x, s, t) \in R^{N + 2}$ . Using the last inequality in (4.1), we obtain $\begin{matrix} (4.2) & \begin{array}{l} \int_{R^{N}} w_{L, n}^{2 (β - 1)} | \nabla w_{n} |^{2} d x + \int_{R^{N}} z_{L, n}^{2 (β - 1)} | \nabla z_{n} |^{2} d x \\ ⩽ C_{1} \int_{R^{N}} w_{n}^{p} w_{L, n}^{2 (β - 1)} d x + C_{1} \int_{R^{N}} z_{n}^{p} z_{L, n}^{2 (β - 1)} d x . \end{array} \end{matrix}$

Let S be the best constant of the embedding $D^{1, 2} (R^{N}) ↪ L^{2^{}} (R^{N})$ and define ${\hat{w}}_{L, n} : = w_{n} w_{L, n}^{β - 1}$ and ${\hat{z}}_{L, n} : = z_{n} z_{L, n}^{β - 1}$ . Since $w_{L, n} ⩽ w_{n}$ and $z_{L, n} ⩽ z_{n}$ , we have that $\begin{matrix} S^{- 1} (‖ {\hat{w}}_{L, n} ‖_{L^{2^{}}}^{2} + ‖ {\hat{z}}_{L, n} ‖_{L^{2^{}}}^{2}) & ⩽ \int_{R^{N}} {| \nabla (w_{n} w_{L, n}^{β - 1}) |}^{2} d x + \int_{R^{N}} {| \nabla (z_{n} z_{L, n}^{β - 1}) |}^{2} d x \\ ⩽ β^{2} \int_{R^{N}} w_{L, n}^{2 (β - 1)} | \nabla w_{n} |^{2} d x + β^{2} \int_{R^{N}} z_{L, n}^{2 (β - 1)} | \nabla z_{n} |^{2} d x . \end{matrix}$

The last inequality and (4.2) provide $\begin{matrix} S^{- 1} (‖ {\hat{w}}_{L, n} ‖_{L^{2^{}}}^{2} + ‖ {\hat{z}}_{L, n} ‖_{L^{2^{}}}^{2}) ⩽ C_{1} β^{2} (\int_{R^{N}} | w_{n} |^{p} w_{L, n}^{2 (β - 1)} d x + \int_{R^{N}} | z_{n} |^{p} z_{L, n}^{2 (β - 1)} d x), \end{matrix}$ for all $β > 1$ .

The above expression, $w_{L, n} ⩽ | w_{n} |$ , $z_{L, n} ⩽ | z_{n} |$ and by definition of ${\hat{w}}_{L, n}$ and ${\hat{z}}_{L, n}$ imply that $\begin{matrix} S^{- 1} (‖ {\hat{w}}_{L, n} ‖_{L^{2^{}}}^{2} + ‖ {\hat{z}}_{L, n} ‖_{L^{2^{}}}^{2}) & ⩽ C_{1} β^{2} \int_{R^{N}} | w_{n} |^{p - 2} | {\hat{w}}_{L, n} |^{2} d x \\ + C_{1} β^{2} \int_{R^{N}} | z_{n} |^{p - 2} | {\hat{z}}_{L, n} |^{2} d x . \end{matrix}$

Using Hölder’s inequality with exponents $\frac{2^{}}{p - 2}$ and $\frac{2^{}}{2^{} - (p - 2)}$ , we get $\begin{matrix} S^{- 1} (‖ {\hat{w}}_{L, n} ‖_{L^{2^{}}}^{2} + ‖ {\hat{z}}_{L, n} ‖_{L^{2^{}}}^{2}) \\ ⩽ C_{1} β^{2} {(\int_{R^{N}} | w_{n} |^{2^{}} d x)}^{(p - 2) / 2^{}} {(\int_{R^{N}} | {\hat{w}}_{L, n} |^{22^{} / (2^{} - (p - 2))} d x)}^{(2^{} - (p - 2)) / 2^{}} \\ + C_{1} β^{2} {(\int_{R^{N}} | z_{n} |^{2^{}} d x)}^{(p - 2) / 2^{}} {(\int_{R^{N}} | {\hat{z}}_{L, n} |^{22^{} / (2^{} - p)} d x)}^{(2^{} - p) / 2^{}}, \end{matrix}$ where $2 < 22^{} / (2^{} - (p - 2)) < 2^{}$ .

Since $(w_{n}, z_{n})$ is bounded in $H^{1} (R^{N}) \times H^{1} (R^{N})$ , we conclude that $\begin{array}{rcl} S^{- 1} (‖ {\hat{w}}_{L, n} ‖_{L^{2^{}}}^{2} + ‖ {\hat{z}}_{L, n} ‖_{L^{2^{}}}^{2}) ⩽ C_{1} β^{2} ‖ {\hat{w}}_{L, n} ‖_{L^{ζ}}^{2} + C_{1} β^{2} ‖ {\hat{z}}_{L, n} ‖_{L^{ζ}}^{2}, \end{array}$ where $ζ = 22^{} / (2^{} - (p - 2))$ .

From Fatou’s lemma in the variable L we obtain $\begin{array}{rcl} ‖ w_{n} ‖_{L^{β 2^{}}} + ‖ z_{n} ‖_{L^{β 2^{}}} & ⩽ & C_{2}^{1 / β} β^{1 / β} ‖ w_{n} ‖_{L^{β ζ}} \\ + C_{2}^{1 / β} β^{1 / β} ‖ z_{n} ‖_{L^{β ζ}}, \end{array}$ whenever $w_{n}^{β ζ}, z_{n}^{β ζ} \in L^{1} (R^{N})$ .

Now, we set $β : = 2^{} / ζ > 1$ and note that, since $w_{n}^{β}, z_{n}^{β} \in L^{ζ} (R^{N})$ , the above inequality holds for this choice of β. Moreover, since $β^{2} ζ = β 2^{}$ , it follows that the inequality also holds with β replaced by $β^{2}$ . Hence, $\begin{array}{rcl} {‖ (w_{n}, z_{n}) ‖}_{L^{β^{2} 2^{}}} ⩽ C_{2}^{1 / β^{2}} β^{2 / β^{2}} {‖ (w_{n}, z_{n}) ‖}_{L^{β^{2} ζ}} . \end{array}$

By iterating this process and recalling that $β ζ = 2^{}$ we obtain, for $k \in N$ , $\begin{matrix} {‖ (w_{n}, z_{n}) ‖}_{L^{β^{k} 2^{}}} ⩽ C_{2}^{\sum_{i = 1}^{k} β^{- i}} β^{\sum_{i = 1}^{m} i β^{- i}} {‖ (w_{n}, z_{n}) ‖}_{L^{2^{}}} . \end{matrix}$ Since $β > 1$ we can take the limit as $k \to \infty$ to get $\begin{matrix} {‖ (w_{n}, z_{n}) ‖}_{L^{\infty} (R^{N})} ⩽ C_{3} {‖ (w_{n}, z_{n}) ‖}_{L^{2^{}}} . \end{matrix}$

From Lemma 4.1, there is $\tilde{K} > 0$ such that $\begin{matrix} {‖ (w_{n}, z_{n}) ‖}_{L^{\infty} (R^{N})} ⩽ \tilde{K}, \end{matrix}$ which prove this lemma. □
Corollary 4.3.
The function $(w_{n}, z_{n})$ converges uniformly on compact sets on $R^{N}$ .
Proof.
Now, for each n fixed, we define $\begin{array}{l} {\tilde{g}}_{n} (x) & = Q_{u} (u_{n} (x + x_{n}), v_{n} (x + x_{n})) - W (ε_{n} x + ε_{n} x_{n}) u_{n} (x + x_{n}) \\ + u_{n} (x + x_{n}) \end{array}$ that satisfies $\begin{matrix} - Δ u_{n} + u_{n} = {\tilde{g}}_{n} (x) a.e in R^{N} \end{matrix}$ and $\begin{matrix} | {\tilde{g}}_{n} (x) | ⩽ (V_{\infty} + 1) u_{n} + C u_{n}^{p - 1} + C v_{n}^{p - 1} a.e in R^{N} . \end{matrix}$

Since $u_{n}, v_{n} \in L^{2} (R^{N}) \cap L^{\infty} (R^{N})$ , from interpolation inequality, we get $u_{n}, v_{n} \in L^{q} (R^{N})$ , $\forall q ⩾ 2$ , that implies ${\tilde{g}}_{n} (x) L^{q} (R^{N})$ , $\forall q ⩾ 2$ . From regularity elliptic theory, we get $u_{n} \in W^{2, q} (R^{N})$ , $\forall q ⩾ 2$ . For q sufficiently large, we obtain $W^{2, q} (R^{N}) ↪ C^{1, α} (R^{N})$ , for some $0 < α < 1$ . Then, $u_{n} \in C^{1, α} (R^{N})$ . Moreover, from Lemma 4.1 and Lemma 4.2, we conclude that $‖ u_{n} ‖_{C_{loc}^{1, α} (R^{N})}$ is bounded. Then, we have $\begin{matrix} u_{n} \to u uniformly on K, \end{matrix}$ for each compact $K \subset R^{N}$ . Repeating the same argument with $v_{n}$ , the proof is over. □
Remark 4.4.
Since the sequence $(x_{n}) \subset R^{N}$ is arbitrary, the regularity obtained in the Corollary 4.3 is true for $(u_{ε}, v_{ε})$ .

5. Proof of Theorem 1.1: The Class 1

In this section, we will prove the Theorem 1.1, by supposing that W and V belongs to Class 1. Consider $\begin{matrix} Ω = B_{R_{ε}} (0), \end{matrix}$ where $R_{ε} = \frac{1}{ε}$ . In what follows, our goal is to prove that there is $ε_{0} > 0$ such that $\begin{matrix} | (u_{ε} (x), v_{ε} (x)) | < α, \forall x \in R^{N} ∖ B_{\frac{R_{ε}}{ε}} (0) and \forall ε \in (0, ε_{0}) . \end{matrix}$

Lemma 5.1.
The function $(u_{ε}, v_{ε})$ verifies the following estimate $\begin{matrix} max_{x \in \partial B_{\frac{R_{ε}}{ε}} (0)} u_{ε} (x) \to 0, ε \to 0 \end{matrix}$ and $\begin{matrix} max_{x \in \partial B_{\frac{R_{ε}}{ε}} (0)} v_{ε} (x) \to 0, ε \to 0 . \end{matrix}$
Proof.
Assume by contradiction that there is $ε_{n} \to 0$ and $γ > 0$ such that $\begin{matrix} max_{x \in \partial B_{\frac{R_{ε_{n}}}{ε_{n}}} (0)} u_{n} (x) ⩾ γ \forall n \in N \end{matrix}$ and $\begin{matrix} max_{x \in \partial B_{\frac{R_{ε_{n}}}{ε_{n}}} (0)} v_{n} (x) ⩾ γ \forall n \in N, \end{matrix}$ where $u_{n} = u_{ε_{n}}$ and $v_{n} = v_{ε_{n}}$ . From now on, we fix $x_{n} \in \partial B_{\frac{R_{ε_{n}}}{ε_{n}}} (0)$ satisfying $\begin{matrix} u_{n} (x_{n}) = max_{x \in \partial B_{\frac{R_{ε_{n}}}{ε_{n}}} (0)} u_{ε_{n}} (x) \end{matrix}$ and $\begin{matrix} v_{n} (x_{n}) = max_{x \in \partial B_{\frac{R_{ε_{n}}}{ε_{n}}} (0)} v_{ε_{n}} (x) . \end{matrix}$ Therefore, $\begin{matrix} u_{n} (x_{n}) ⩾ γ, \forall n \in N \end{matrix}$ and $\begin{matrix} v_{n} (x_{n}) ⩾ γ, \forall n \in N . \end{matrix}$

By Lemma 4.1, $(u_{n}, v_{n})$ is bounded in $H^{1} (R^{N}) \times H^{1} (R^{N})$ . Thereby, setting $(w_{n} = u_{n} (\cdot + x_{n}), z_{n} = v_{n} (\cdot + x_{n}))$ , we can guarantee that $(w_{n}, z_{n})$ is also bounded in $H^{1} (R^{N}) \times H^{1} (R^{N})$ and it satisfies $\begin{matrix} \{\begin{matrix} - Δ w_{n} + W (ε_{n} x + ε_{n} x_{n}) w_{n} = H_{w} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}), x \in R^{N}, \\ - Δ z_{n} + V (ε_{n} x + ε_{n} x_{n}) z_{n} = H_{z} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}), x \in R^{N}, \\ w_{n}, z_{n} \in H^{1} (R^{N}) . \end{matrix} \end{matrix}$

From Corollary 4.3, $(w_{n}, z_{n})$ converges uniformly on compact set on $R^{N}$ for its weak limit $(w, z) \in H^{1} (R^{N}) \times H^{1} (R^{N})$ . Then, $(w, z) \in C (R^{N}) \times C (R^{N})$ and $w (0) ⩾ γ$ , $z (0) ⩾ γ$ , implying that $w, z \neq 0$ . Moreover, by $(A_{2})$ , there is a subsequence of $(ε_{n} x_{n})$ , still denote by itself, such that $\begin{matrix} α_{1} = lim_{n \to + \infty} W (ε_{n} x_{n}) \end{matrix}$ and $\begin{matrix} α_{2} = lim_{n \to + \infty} V (ε_{n} x_{n}), \end{matrix}$ for some $α_{1} > 0$ , $α_{2} > 0$ . Since for each $ϕ, ψ \in H^{1} (R^{N})$ , the equality below holds $\begin{array}{l} \int_{R^{N}} \nabla w_{n} \nabla ϕ d x + \int_{R^{N}} W (ε_{n} x + ε_{n} x_{n}) w_{n} ϕ d x \\ - \int_{R^{N}} H_{w} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}) ϕ d x = o_{n} (1) ‖ ϕ ‖ \end{array}$ and $\begin{array}{l} \int_{R^{N}} \nabla z_{n} \nabla ψ d x + \int_{R^{N}} V (ε_{n} x + ε_{n} x_{n}) z_{n} ψ d x \\ - \int_{R^{N}} H_{z} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}) ψ d x = o_{n} (1) ‖ ψ ‖ . \end{array}$

Taking the limit of $n \to + \infty$ , let us deduce that $(w, z)$ is a nontrivial solution of the problem $\begin{matrix} (5.1) & \begin{array}{l} Δ w - α_{1} w + H_{w} (x, w, z) = 0, x \in R^{N}, \\ Δ z - α_{2} z + H_{z} (x, w, z) = 0, x \in R^{N}, \end{array} \end{matrix}$ where $\begin{matrix} H (x, s, t) : = I_{Λ} (x) Q (s, t) + (1 - I_{Λ} (x)) \hat{Q} (s, t), \end{matrix}$ for some $I_{Λ} \in L^{\infty} (R^{N})$ . Thus, by regularity theory, $(w, z) \in L^{\infty} (R^{N}) \cap H^{2} (R^{N}) \times L^{\infty} (R^{N}) \cap H^{2} (R^{N})$ .

For each $j \in N$ , there is $ϕ_{j}, ψ_{j} \in C_{0}^{\infty} (R^{N})$ such that $\begin{matrix} ‖ ϕ_{j} - w ‖ ⩽ 1 / j and ‖ ψ_{j} - z ‖ ⩽ 1 / j, \end{matrix}$ that is, $\begin{matrix} ‖ ϕ_{j} - w ‖ = o_{j} (1) and ‖ ψ_{j} - z ‖ = o_{j} (1) . \end{matrix}$

Using $(\frac{\partial ϕ_{j}}{\partial x_{i}}, \frac{\partial ψ_{j}}{\partial x_{i}})$ as a test function, we get $\begin{array}{l} \int_{R^{N}} \nabla w_{n} \nabla \frac{\partial ϕ_{j}}{\partial x_{i}} d x + \int_{R^{N}} W (ε_{n} x + ε_{n} x_{n}) w_{n} \frac{\partial ϕ_{j}}{\partial x_{i}} d x \\ - \int_{R^{N}} H_{w} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}) \frac{\partial ϕ_{j}}{\partial x_{i}} d x + \int_{R^{N}} \nabla z_{n} \nabla \frac{\partial ψ_{j}}{\partial x_{i}} d x \\ + \int_{R^{N}} V (ε_{n} x + ε_{n} x_{n}) z_{n} \frac{\partial ψ_{j}}{\partial x_{i}} d x - \int_{R^{N}} H_{z} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}) \frac{\partial ψ_{j}}{\partial x_{i}} d x = o_{n} (1) . \end{array}$

Now, using well known arguments, $\begin{array}{l} \int_{R^{N}} \nabla w_{n} \nabla \frac{\partial ϕ_{j}}{\partial x_{i}} d x = \int_{R^{N}} \nabla w \nabla \frac{\partial ϕ_{j}}{\partial x_{i}} d x + o_{n} (1), \\ \int_{R^{N}} \nabla z_{n} \nabla \frac{\partial ψ_{j}}{\partial x_{i}} d x = \int_{R^{N}} \nabla z \nabla \frac{\partial ψ_{j}}{\partial x_{i}} d x + o_{n} (1), \\ \int_{R^{N}} H_{w} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}) \frac{\partial ϕ_{j}}{\partial x_{i}} d x = \int_{R^{N}} H_{w} (x, w, z) \frac{\partial ϕ_{j}}{\partial x_{i}} d x + o_{n} (1) \end{array}$ and $\begin{matrix} \int_{R^{N}} H_{z} (ε_{n} x + ε_{n} x_{n}, w_{n}, z_{n}) \frac{\partial ψ_{j}}{\partial x_{i}} d x = \int_{R^{N}} H_{z} (x, w, z) \frac{\partial ϕ_{j}}{\partial x_{i}} d x + o_{n} (1) . \end{matrix}$

Gathering the above limit with (5.1), we find $\begin{matrix} \underset{n \to + \infty}{lim sup} | \int_{R^{N}} (W (ε_{n} x + ε_{n} x_{n}) - W (ε_{n} x_{n})) w_{n} \frac{\partial ϕ_{j}}{\partial x_{i}} d x | = 0 \end{matrix}$ and $\begin{matrix} \underset{n \to + \infty}{lim sup} | \int_{R^{N}} (V (ε_{n} x + ε_{n} x_{n}) - V (ε_{n} x_{n})) z_{n} \frac{\partial ψ_{j}}{\partial x_{i}} d x | = 0 . \end{matrix}$

As $ϕ_{j}$ , $ψ_{j}$ has compact support, the above limit gives $\begin{matrix} \underset{n \to + \infty}{lim sup} | \int_{R^{N}} (W (ε_{n} x + ε_{n} x_{n}) - W (ε_{n} x_{n})) w \frac{\partial ϕ_{j}}{\partial x_{i}} d x | = 0 \end{matrix}$ and $\begin{matrix} \underset{n \to + \infty}{lim sup} | \int_{R^{N}} (V (ε_{n} x + ε_{n} x_{n}) - V (ε_{n} x_{n})) z \frac{\partial ψ_{j}}{\partial x_{i}} d x | = 0 . \end{matrix}$

Now, recalling that $\frac{\partial w}{\partial x_{i}}, \frac{\partial z}{\partial x_{i}} \in L^{2} (R^{N})$ , we have that $(\frac{\partial ϕ_{j}}{\partial x_{i}}, \frac{\partial ψ_{j}}{\partial x_{i}})$ is bounded in $L^{2} (R^{N}) \times L^{2} (R^{N})$ . Hence, $\begin{matrix} \underset{n \to + \infty}{lim sup} | \int_{R^{N}} (W (ε_{n} x + ε_{n} x_{n}) - W (ε_{n} x_{n})) ϕ_{j} \frac{\partial ϕ_{j}}{\partial x_{i}} d x | = o_{j} (1) \end{matrix}$ and $\begin{matrix} \underset{n \to + \infty}{lim sup} | \int_{R^{N}} (V (ε_{n} x + ε_{n} x_{n}) - V (ε_{n} x_{n})) ψ_{j} \frac{\partial ψ_{j}}{\partial x_{i}} d x | = o_{j} (1) . \end{matrix}$

Then, $\begin{matrix} \underset{n \to + \infty}{lim sup} | \frac{1}{2} \int_{R^{N}} (W (ε_{n} x + ε_{n} x_{n}) - W (ε_{n} x_{n})) \frac{\partial (ϕ_{j}^{2})}{\partial x_{i}} d x | = o_{j} (1) \end{matrix}$ and $\begin{matrix} \underset{n \to + \infty}{lim sup} | \frac{1}{2} \int_{R^{N}} (V (ε_{n} x + ε_{n} x_{n}) - V (ε_{n} x_{n})) \frac{\partial (ψ_{j}^{2})}{\partial x_{i}} d x | = o_{j} (1) . \end{matrix}$

Using Green’s Theorem together with the fact that $(ϕ_{j}, ψ_{j})$ has compact support, we obtain the limit below $\begin{matrix} \underset{n \to + \infty}{lim sup} | \int_{R^{N}} \frac{\partial W}{\partial x_{i}} (ε_{n} x + ε_{n} x_{n}) ϕ_{j}^{2} d x | = o_{j} (1) \end{matrix}$ and $\begin{matrix} \underset{n \to + \infty}{lim sup} | \int_{R^{N}} \frac{\partial V}{\partial x_{i}} (ε_{n} x + ε_{n} x_{n}) ψ_{j}^{2} d x | = o_{j} (1), \end{matrix}$ which combined with $(A_{1})$ loads to $\begin{matrix} \underset{n \to + \infty}{lim sup} | \frac{\partial W}{\partial x_{i}} (ε_{n} x_{n}) \int_{R^{N}} | ϕ_{j} |^{2} d x | = o_{j} (1) \end{matrix}$ and $\begin{matrix} \underset{n \to + \infty}{lim sup} | \frac{\partial V}{\partial x_{i}} (ε_{n} x_{n}) \int_{R^{N}} | ψ_{j} |^{2} d x | = o_{j} (1) . \end{matrix}$ As $\begin{matrix} \int_{R^{N}} | ϕ_{j} |^{2} d x \to \int_{R^{N}} | w |^{2} d x > 0 as j \to + \infty \end{matrix}$ and $\begin{matrix} \int_{R^{N}} | ψ_{j} |^{2} d x \to \int_{R^{N}} | z |^{2} d x > 0 as j \to + \infty, \end{matrix}$ it follows that $\begin{matrix} \underset{n \to + \infty}{lim sup} | \frac{\partial W}{\partial x_{i}} (ε_{n} x_{n}) | = o_{j} (1), \forall i \in {1, \dots, N} \end{matrix}$ and $\begin{matrix} \underset{n \to + \infty}{lim sup} | \frac{\partial V}{\partial x_{i}} (ε_{n} x_{n}) | = o_{j} (1), \forall i \in {1, \dots, N} . \end{matrix}$ Since j is arbitrary, we derive that $\begin{matrix} \nabla W (ε_{n} x_{n}) \to 0 as n \to \infty \end{matrix}$ and $\begin{matrix} \nabla V (ε_{n} x_{n}) \to 0 as n \to \infty . \end{matrix}$

Therefore, $(ε_{n} x_{n})$ is a ${(PS)}_{α_{1}, α_{2}}$ sequence for W and V, which is an absurd, since by $(A_{2})$ , W and V satisfies the $(PS)$ condition and $(ε_{n} x_{n})$ does not have any convergent subsequence in $R^{N}$ , because $\begin{matrix} | ε_{n} x_{ε_{n}} | = R_{ε_{n}} = \frac{1}{ε_{n}} \to + \infty, as n \to + \infty . \end{matrix}$ □
Proof of Theorem 1.1 (conclusion).
From Lemma 5.1, there is $ε_{0} > 0$ such that $\begin{matrix} max_{x \in \partial B_{\frac{R_{ε}}{ε}} (0)} u_{ε} (x) < α, \forall ε \in (0, ε_{0}) \end{matrix}$ and $\begin{matrix} max_{x \in \partial B_{\frac{R_{ε}}{ε}} (0)} v_{ε} (x) < α, \forall ε \in (0, ε_{0}) . \end{matrix}$

Considering the functions $\begin{matrix} {\tilde{u}}_{ε} (x) = \{\begin{matrix} 0, & x \in {\overline{B}}_{\frac{R_{ε}}{ε}} (0), \\ {(u_{ε} - α)}^{+} (x), & x \in R^{N} ∖ B_{\frac{R_{ε}}{ε}} (0) \end{matrix} \end{matrix}$ and $\begin{matrix} {\tilde{v}}_{ε} (x) = \{\begin{matrix} 0, & x \in {\overline{B}}_{\frac{R_{ε}}{ε}} (0), \\ {(v_{ε} - α)}^{+} (x), & x \in R^{N} ∖ B_{\frac{R_{ε}}{ε}} (0), \end{matrix} \end{matrix}$ it follows that ${\tilde{u}}_{ε}, {\tilde{v}}_{ε} \in H^{1} (R^{N})$ . Thereby, $J_{ε}^{'} (u_{ε}, v_{ε}) ({\tilde{u}}_{ε}, {\tilde{v}}_{ε}) = 0$ , or equivalently, $\begin{array}{l} \int_{R^{N}} \nabla u_{ε} \nabla {\tilde{u}}_{ε} d x + \int_{R^{N}} W (ε x) u_{ε} {\tilde{u}}_{ε} d x + \int_{R^{N}} \nabla v_{ε} \nabla {\tilde{v}}_{ε} d x + \int_{R^{N}} V (ε x) v_{ε} {\tilde{v}}_{ε} d x \\ = \int_{R^{N}} H_{u} (x, u_{ε}, v_{ε}) {\tilde{u}}_{ε} d x + \int_{R^{N}} H_{v} (x, u_{ε}, v_{ε}) {\tilde{v}}_{ε} d x . \end{array}$

Now, using the definition of H, it is possible to prove that ${\tilde{u}}_{ε} \equiv 0$ and ${\tilde{v}}_{ε} \equiv 0$ . From this, $\begin{matrix} u_{ε} (x) ⩽ α, \forall x \in R^{N} ∖ B_{\frac{R_{ε}}{ε}} (0) \end{matrix}$ and $\begin{matrix} v_{ε} (x) ⩽ α, \forall x \in R^{N} ∖ B_{\frac{R_{ε}}{ε}} (0), \end{matrix}$ showing that $(u_{ε}, v_{ε})$ is a solution for $(\hat{S_{ε}})$ . □

6. Proof of Theorem 1.1: The Class 2

In this section, we will prove the Theorem 1.1 for the case that W and V belongs to Class 2. However, we will use the results showed in Section 2 with $\begin{matrix} Ω = Λ . \end{matrix}$

Next, we will show that there is $ε_{0} > 0$ such that $\begin{matrix} | (u_{ε} (x), v_{ε} (x)) | < α, \forall x \in R^{N} ∖ Λ_{ε} and \forall ε \in (0, ε_{0}) . \end{matrix}$

Lemma 6.1.

The function $(u_{ε}, v_{ε})$ verifies the following estimate $\begin{matrix} max_{x \in \partial Λ_{ε}} u_{ε} (x) \to 0, ε \to 0 \end{matrix}$ and $\begin{matrix} max_{x \in \partial Λ_{ε}} v_{ε} (x) \to 0, ε \to 0 . \end{matrix}$

Proof.

Using the same type of arguments explored in the proof of Lemma 5.1, we find a sequence $(x_{n}) \subset \partial Λ_{ε_{n}}$ , with $ε_{n} \to 0$ , satisfying $\begin{matrix} \nabla W (ε_{n} x_{n}) \to 0 \end{matrix}$ and $\begin{matrix} \nabla V (ε_{n} x_{n}) \to 0 . \end{matrix}$

Since $(ε_{n} x_{n}) \subset \partial Λ$ , and $\partial Λ$ is a compact set in $R^{N}$ , we can assume that there is $x_{0} \in \partial Λ$ such that $\begin{matrix} ε_{n} x_{n} \to x_{0} in R^{N} . \end{matrix}$

Gathering the above limits with the fact that $W, V \in C^{1} (R^{N}, R)$ , we get $\begin{matrix} x_{0} \in \partial Λ and \nabla W (x_{0}) = 0 \end{matrix}$ and $\begin{matrix} x_{0} \in \partial Λ and \nabla V (x_{0}) = 0, \end{matrix}$ contradicting $(A_{3})$ . □

Proof of Theorem 1.1 (conclusion).

The conclusion of the proof follows as in Section 5. □

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Existence and asymptotic behavior of solutions for a class of semilinear subcritical elliptic systems

Abstract

Keywords

1. Introduction

Proof. The proof is a consequence of Lemma 3.2, Lemma 3.3 and Mountain Pass Theorem [5]. □ 4. Some uniform estimates

References

Proof.
The proof is a consequence of Lemma 3.2, Lemma 3.3 and Mountain Pass Theorem [5]. □
4. Some uniform estimates