- Research Article
- Open Access

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

- Guoqing Zhang
^{1}Email author and - Sanyang Liu
^{2}

**2008**:217636

https://doi.org/10.1155/2008/217636

© G. Zhang and S. Liu. 2008

**Received:**31 October 2007**Accepted:**4 March 2008**Published:**12 March 2008

## Abstract

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.

## Keywords

- Weak Solution
- Elliptic Problem
- Nontrivial Solution
- Mountain Pass
- Sobolev Embedding

## 1. Introduction

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

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

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

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

Here, we assume the following condition:

(H1)

(H2) uniformly with respect to

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

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

Moreover,

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,

Deffinition.

## 3. Proof of Theorem 2.2

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

Proof.

Lemma 3.2.

There exist and such that

and so, taking large, we get

Proof of Theorem 2.2.

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.

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

Claim 3.4.

which is a contradiction to as

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

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

## References

- Colin F, Frigon M:
**Systems of coupled Poisson equations with critical growth in unbounded domains.***Nonlinear Differential Equations and Applications*2006,**13**(3):369-384. 10.1007/s00030-006-4012-1MathSciNetView ArticleMATHGoogle Scholar - Ding Y, Li S:
**Existence of entire solutions for some elliptic systems.***Bulletin of the Australian Mathematical Society*1994,**50**(3):501-519. 10.1017/S0004972700013605MathSciNetView ArticleMATHGoogle Scholar - de Figueiredo DG:
**Nonlinear elliptic systems.***Anais da Academia Brasileira de Ciências*2000,**72**(4):453-469. 10.1590/S0001-37652000000400002MathSciNetView ArticleGoogle Scholar - de Figueiredo DG, do Ó JM, Ruf B:
**Critical and subcritical elliptic systems in dimension two.***Indiana University Mathematics Journal*2004,**53**(4):1037-1054. 10.1512/iumj.2004.53.2402MathSciNetView ArticleMATHGoogle Scholar - de Figueiredo DG, Yang J:
**Decay, symmetry and existence of solutions of semilinear elliptic systems.***Nonlinear Analysis: Theory, Methods & Applications*1998,**33**(3):211-234. 10.1016/S0362-546X(97)00548-8MathSciNetView ArticleMATHGoogle Scholar - Li G, Yang J:
**Asymptotically linear elliptic systems.***Communications in Partial Differential Equations*2004,**29**(5-6):925-954.MathSciNetView ArticleMATHGoogle Scholar - Adimurthi , Yadava SL:
**Multiplicity results for semilinear elliptic equations in a bounded domain of****involving critical exponents.***Annali della Scuola Normale Superiore di Pisa*1990,**17**(4):481-504.MathSciNetMATHGoogle Scholar - de Figueiredo DG, Miyagaki OH, Ruf B:
**Elliptic equations in****with nonlinearities in the critical growth range.***Calculus of Variations and Partial Differential Equations*1995,**3**(2):139-153. 10.1007/BF01205003MathSciNetView ArticleMATHGoogle Scholar - Shen Y, Yao Y, Chen Z:
**On a class of nonlinear elliptic problem with critical potential in**.*Science in China Series A*2004,**34:**610-624.MathSciNetGoogle Scholar - Cao DM:
**Nontrivial solution of semilinear elliptic equation with critical exponent in**.*Communications in Partial Differential Equations*1992,**17**(3-4):407-435. 10.1080/03605309208820848MathSciNetView ArticleMATHGoogle Scholar - Alves CO, do Ó JM, Miyagaki OH:
**On nonlinear perturbations of a periodic elliptic problem in****involving critical growth.***Nonlinear Analysis: Theory, Methods & Applications*2004,**56**(5):781-791. 10.1016/j.na.2003.06.003MathSciNetView ArticleMATHGoogle Scholar - Kryszewski W, Szulkin A:
**Generalized linking theorem with an application to a semilinear Schrödinger equation.***Advances in Differential Equations*1998,**3**(3):441-472.MathSciNetMATHGoogle Scholar - Willem M:
*Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications*.*Volume 24*. Birkhäuser, Boston, Mass, USA; 1996:x+162.Google Scholar - Li G, Szulkin A:
**An asymptotically periodic Schrödinger equation with indefinite linear part.***Communications in Contemporary Mathematics*2002,**4**(4):763-776. 10.1142/S0219199702000853MathSciNetView ArticleMATHGoogle Scholar - do Ó JM, Souto MAS:
**On a class of nonlinear Schrödinger equations in****involving critical growth.***Journal of Differential Equations*2001,**174**(2):289-311. 10.1006/jdeq.2000.3946MathSciNetView ArticleMATHGoogle Scholar

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