1. Introduction

In this paper, we consider the following systems with an arbitrary number of coupled Poisson equations with critical growth in punctured unbounded domains . More precisely, we study the following system: where , , , for all , and where , using the following notations: , and . We will show in Section 3 that the corresponding variational formulation is given by

The first difficulty to be tackled in order to prove the existence of a solution for the system is due to the fact that the preceding functional has a strong indefinite quadratic part. Therefore the classical min-max results cannot be applied, unlike the generalized linking theorem presented by Kryszewski and Szulkin (cf. [1]).

Let us mention that similar types of problems for systems of two equations on bounded domains were studied in the subcritical growth case by Husholf and van der Vorst [2] using the Indefinite Functional Theorem due to Benci and Rabinowitz [3], and by Felmer and Wang [4] who obtained multiplicity results in using Galerkin type methods. The critical growth case was studied by Husholf et al. [5] where they used a dual formulation due to Clarke and Ekeland [6]. Let us also mention that Zhang and Liu in [7], and Alves et al. in [8], studied elliptic systems with two equations on unbounded domains, while Silva and Xavier [9] showed the existence of multiple solutions for similar systems on smooth bounded domains, where the Laplacian is replaced with the -Laplacian.

In 2005, Frigon and the author studied in [10] the following system of two coupled Poisson equations with critical growth:

on unbounded punctured domains of the form

where and where is a bounded domain with a -boundary, such that , . Indeed, these domains are invariant under -translations and have a -boundary. Due to the latter invariance and the periodicity of the function , the corresponding functional, besides its strong indefinite quadratic part, is also invariant under -translations. Consequently, the Palais-Smale condition fails at every critical level, and that is a second difficulty to overcome. In the proof of the existence of a solution for the above system, the generalized linking theorem of Krysewki and Szulkin was used to obtain a Palais-Smale sequence, and a concentration-compactness lemma à la Lions also proved in [10] was invoked in order to show the nontriviality of the solution. The same method will be applied to the system .

In addition, we would like to fulfill the following classical assumptions.

where , (resp., ), denotes the gradient of with respect to the variables , (resp., ).

The main result of this paper can now be stated as follows.

Theorem 1.1 (existence of a nontrivial solution). Let be a punctured domain defined in (1.3), and let be a function satisfying assumptions , , and . Then the system has a nontrivial solution.

This paper is organized as follows. In the next section, the key concentration-compactness lemma, and the generalized linking of Krysewski and Szulkin are presented. In Section 3, we show that the functional fulfills the assumptions of the Krysewski and Szulkin theorem. The proof of the main theorem is presented in Section 4. Finally, Section 5 is devoted to an application of the main result to reaction-diffusion systems with skew-gradient structure. Existence of steady-states solutions for these systems will be established.

2. Preliminaries

In the sequel, we will study the case , and let . Let us define the Hilbert space

endowed with the inner product

and the associated norm noted . The Sobolev Imbedding Theorem asserts that the imbedding

is continuous.

For a domain , we denote by the closure of in . Obviously , and , provided that the Poincaré Inequality is satisfied.

2.1. Concentration-Compactness Lemmas

Let us recall that the embedding is not compact because of the action of dilatations, but we have the following well-known result (refer to Wang and Willem [11] for a generalization).

Lemma 2.1. If in , then in .

The following lemma due to Ramos, Ramos et al. [12] gives sufficient conditions ensuring the convergence to in of a sequence in . This type of results was firstly established by Lions [13] for an exponent . See also Colin [14] or [15] for a similar result in a weighted space on a cylindrical domain.

Lemma 2.2. Let . If is bounded in and if then in .

In [10], Frigon and the author proved a similar result for punctured unbounded domains invariant by -translations. As mentioned earlier, this lemma will be used in order to prove the nontriviality of the weak solution given by the Krysewski-Szulkin's theorem.

Lemma 2.3. Let be a bounded sequence, where is a punctured unbounded domain defined as in (1.3). If then in

2.2. Kryszewski-Szulkin Linking Theorem

In 1996, Krysewski and Szulkin [1] (interested readers could also refer to [16] for an elegant proof) presented a generalized linking theorem for a suitable functional defined on a Hilbert space with a separable subspace of which could be infinite dimensional, and . Let us state a corollary of their result that will be sufficient for our purposes.

Let be the orthogonal projections. Now, let and let be such that Define

Theorem 2.4 (see [1]). Let be weakly sequentially lower semicontinuous, bounded below and such that is weakly sequentially continuous. If satisfies then there exists and a sequence such that

3. Existence of a Bounded Palais-Smale Sequence for the System

For the sake of simplicity and readability, we chose to divide the present section into five subsections, each of them dedicated to a specific aspect of the assumptions that must be fulfilled by the functional in order to apply the Krysewski-Szulkin theorem.

3.1. Functional Setting

First of all, we establish some general results. Let be a punctured domain in defined as in (1.3). Denote by , the Hilbert space endowed with the inner product

where

and where stands for the gradient operator with respect to . In the sequel, the norm induced by the preceding inner product on the space will be denoted by . If we set

then . Let us denote by (resp., ) the projection of onto (resp., ) and let us define the functional by

where

and where satisfies the assumptions , , and . We will see in Section 3.3 (Lemma 3.3) that the system allows a variational formulation since its solutions will correspond to critical points of in .

3.2. Growth Conditions on the Function

The assumptions , , and have important consequences on the growth of , that are summarized in the next two lemmas.

Lemma 3.1. Under assumption , there exists such that

Proof. We have

The next growth condition is almost identical to the one considered by several authors (see e.g., [17, 18] or [19]).

Lemma 3.2. If satisfies and then there exist constants such that

Proof. Let be the set given by , and let us define, for every , the function as Next, Letting , we get from the periodicity of (assumption ), and from that . An integration of inequality (3.10) results in Now, let us consider such that . Next, let , , and , so we have . Inequality (3.11) implies that Since by the continuity of (assumption ) we have then there exists such that Finally, the result holds for all because of the periodicity of the function .

3.3. Regularity of the Functional

Lemma 3.3. The function is . Moreover, for every ,

Proof. Let . For and , there exists such that On the other hand, assumption implies specific growth conditions on and that lead to Using the Hölder inequality, we conclude that the term on the right-hand side is in , since . Hence, the Lebesgue dominated convergence theorem implies that Now, assume that in . From Lemma 2.1 and assumption , we deduce a.e. in . The continuous embedding and assumption imply that are, respectively, bounded in the spaces and for every such that because of the continuous embedding . Now a direct application of the Lebesgue’s Dominated Convergence Theorem gives the continuity of the Gâteau derivative of and hence is .
On the other hand,
This Gâteau derivative is obviously continuous; so is and

Lemma 3.4. Under assumptions and , the map is weakly sequentially lower semicontinuous, while the map is weakly sequentially continuous.

Proof. Suppose that in . So, and are bounded in , and consequently in . Lemma 2.1 implies that and in . Going, if necessary to a subsequence, and a.e. on , thus a.e. on , by the continuity of the function . Now, the growth condition (3.6) together with the Fatou's lemma implies that On the other hand, reusing the arguments presented in the preceding proof, we have for every , that is, . Moreover, is bounded in , so .

3.4. The Functional Has a Linking Geometry

Choose such that , and , where is the ball with center at the origin, and radius . Let and be defined, respectively, by (2.6) and (2.7).

Lemma 3.5. There exists such that Moreover, there exists such that

Proof. The Sobolev Imbedding Theorem of implies directly (3.26) since for , Next, observe that on we have On the other hand, invoking Lemma 3.2, we have on Since the function has a compact support in , it follows that for some positive constants and . Therefore we have, for thanks to the inequality . Thus, for some ,
Finally, the Cauchy-Schwarz inequality and the Sobolev inequality imply that maps bounded sets into bounded sets, hence

3.5. Boundness of the Palais-Smale Sequence

Boundness of the Palais-Smale sequence implies the existence of a limit for a convenient subsequence, with respect to the weak topology.

Lemma 3.6. There exists and a bounded sequence in such that

Proof. It follows from Theorem 2.4 and Lemmas 3.3–3.5 that there exist and a sequence in satisfying (3.33).
Let . Observe that for large enough, assumption and (3.33) lead to
Since , there exists such that