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Existence Result for a Class of Elliptic Systems with Indefinite Weights in


We obtain the existence of a nontrivial solution for a class of subcritical elliptic systems with indefinite weights in . The proofs base on Trudinger-Moser inequality and a generalized linking theorem introduced by Kryszewski and Szulkin.

1. Introduction

In this paper, we study the existence of a nontrivial solution for the following systems of two semilinear coupled Poisson equations


where and are continuous functions on and have the maximal growth on which allows to treat problem (P) variationally, is the Laplace operator.

Recently, there exists an extensive bibliography in the study of elliptic problem in [16]. As dimensions in 1998, de Figueiredo and Yang [5] considered the following coupled elliptic systems:


where are radially symmetric in and satisfied the following Ambrosetti-Rabinowitz condition:


and for some . They obtained the decay, symmetry, and existence of solutions for problem (1.2). In 2004, Li and Yang [6] proved that problem (1.2) possesses at least a positive solution when the nonlinearities and are "asymptotically linear" at infinity and "superlinear" at zero, that is,

(1) uniformly in

(2) uniformly with respect to

In 2006, Colin and Frigon [1] studied the following systems of coupled Poission equations with critical growth in unbounded domains:


where is critical Sobolev exponent, and with for a domain containing the origin Here, denotes the open ball centered at the origin of radius . The existence of a nontrivial solution was obtained by using a generalized linking theorem.

As it is well known in dimensions the nonlinearities are required to have polynomial growth at infinity, so that one can define associated functionals in Sobolev spaces. Coming to dimension much faster growth is allowed for the nonlinearity. In fact, the Trudinger-Moser estimates in replace the Sobolev embedding theorem used in

In dimension Adimurth and Yadava [7], de Figueiredo et al. [8] discussed the solvability of problems of the type


where is some bounded domain in Shen et al. [9] considered the following nonlinear elliptic problems with critical potential:


and obtained some existence results. In the whole space some authors considered the following single semilinear elliptic equations:


As the potential and the nonlinearity are asymptotic to a constant function, Cao [10] obtained the existence of a nontrivial solution. As the potential and the nonlinearity are asymptotically periodic at infinity, Alves et al. [11] proved the existence of at least one positive weak solution.

Our aim in this paper is to establish the existence of a nontrivial solution for problem (P) in subcritical case. To our knowledge, there are no results in the literature establishing the existence of solutions to these problems in the whole space. However, it contains a basic difficulty. Namely, the energy functional associated with problem (P) has strong indefinite quadratic part, so there is not any more mountain pass structure but linking one. Therefore, the proofs of our main results cannot rely on classical min-max results. Combining a generalized linking theorem introduced by Kryszewski and Szulkin [12] and Trudinger-Moser inequality, we prove an existence result for problem (P).

The paper is organized as follows. In Section 2, we recall some results and state our main results. In Section 3, main result is proved.

2. Preliminaries and Main Results

Consider the Hilbert space [13]


and denote the product space endowed with the inner product:


If we define


It is easy to check that since


Let us denote by (resp., ) the projection of on to (resp., ), we have


Now, we define the functional




Let and let we define


Here, we assume the following condition:


(H2) uniformly with respect to

(H3) there exist and such that


Lemma 2.1 (see [12, 14]).

Assume (H1), (H2), and (H3), and suppose

(1) where is sequentially lower semicontinu- ous, bounded below, and is weakly sequentially continuous;

(2)there exist , and , such that


Then, there exist and a sequence such that



Theorem 2.2.

Under the assumptions (H1), (H2), and (H3), if and has subcritical growth (see definition below), problem (P) possesses a nontrivial weak solution.

In the whole space do Ó and Souto [15] proved a version of Trudinger-Moser inequality, that is,

(i)if , we have


(ii)if and then there exists a constant such that



We say has subcritical growth at if for all there exists a positive constant such that


3. Proof of Theorem 2.2

In this section, we will prove Theorem 2.2. under our assumptions and (2.14), there exist such that


Then, we obtain


Therefore, the functional is well defined. Furthermore, using standard arguments, we obtain the functional is functional in and


Consequently, the weak solutions of problem (P) are exactly the critical points of in Now, we prove that the functional satisfied the geometry of Lemma 2.1.

Lemma 3.1.

There exist and such that


By (2.14) and assumption (H2), there exists such that


and thus on we have


So, by the Sobolev embedding theorem and (2.12), we can choose sufficiently small, such that


Lemma 3.2.

There exist and such that


  1. (1)

    By assumption (H3), we have on


because for any

  1. (2)

    Assumption (H3) implies that there exist such that


Now, we choose such that , then


Because it follows that for


and so, taking large, we get

Proof of Theorem 2.2.

By Lemma 3.1, there exist and such that By Lemma 3.2, there exist and such that Since , we have


From (2.14), (3.1), and assumption (H3), and is sequentially lower semicontinuous by and Fatou's lemma; is weakly sequentially continuous. Thus, by Lemma 2.1 there exists a sequence such that


Claim 3.3.

There is such that for any Indeed, from (3.12), we obtain that the sequence satisfies


where as Taking in (3.13) and assumption (H3), we have


where depends only on and in assumption (H3). Since we have and thus


On the other hand, let in (3.13), we obtain


that is,


Now, we recall the following inequality (see [7, Lemma 2.4]):


Let and where is defined in (2.14), we have


By (2.12), we have By (2.14), we have


Hence, we have


for some positive constant So we have


Using a similar argument, we obtain


for some positive constant Combining (3.22) and (3.23), we have


for some positive constant which implies that . Thus, for a subsequence still denoted by there is such that


Then, there exists such that From (2.12) and (2.14), we have this implies


Similarly, we can obtain


From these, we have , so is weak solution of problem (P).

Claim 3.4.

is nontrivial. By contradiction, since has subcritical growth, from (2.14) and Hölder inequality, we have


where Choosing suitable and we have


Then, we obtain


Since in as this will lead to


Similarly, we have


Using assumption (H3), we obtain


This together with we have


Thus, we see that


which is a contradiction to as

Consequently, we have a nontrivial critical point of the functional and conclude the proof of Theorem 2.2.


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This work is supported by Innovation Program of Shanghai Municipal Education Commission under Grant no. 08 YZ93.

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Correspondence to Guoqing Zhang.

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Zhang, G., Liu, S. Existence Result for a Class of Elliptic Systems with Indefinite Weights in . Bound Value Probl 2008, 217636 (2008).

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