 Research Article
 Open Access
 Published:
Existence Result for a Class of Elliptic Systems with Indefinite Weights in
Boundary Value Problems volume 2008, Article number: 217636 (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 TrudingerMoser 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 [1–6]. As dimensions in 1998, de Figueiredo and Yang [5] considered the following coupled elliptic systems:
where are radially symmetric in and satisfied the following AmbrosettiRabinowitz 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 TrudingerMoser 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 minmax results. Combining a generalized linking theorem introduced by Kryszewski and Szulkin [12] and TrudingerMoser 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
where
Let and let we define
Here, we assume the following condition:
(H1)
(H2) uniformly with respect to
(H3) there exist and such that
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
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 TrudingerMoser inequality, that is,
(i)if , we have
(ii)if and then there exists a constant such that
Deffinition.
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
Proof.
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
Proof.

(1)
By assumption (H3), we have on
(3.7)
because for any

(2)
Assumption (H3) implies that there exist such that
(3.8)
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.
References
 1.
Colin F, Frigon M: Systems of coupled Poisson equations with critical growth in unbounded domains. Nonlinear Differential Equations and Applications 2006,13(3):369384. 10.1007/s0003000640121
 2.
Ding Y, Li S: Existence of entire solutions for some elliptic systems. Bulletin of the Australian Mathematical Society 1994,50(3):501519. 10.1017/S0004972700013605
 3.
de Figueiredo DG: Nonlinear elliptic systems. Anais da Academia Brasileira de Ciências 2000,72(4):453469. 10.1590/S000137652000000400002
 4.
de Figueiredo DG, do Ó JM, Ruf B: Critical and subcritical elliptic systems in dimension two. Indiana University Mathematics Journal 2004,53(4):10371054. 10.1512/iumj.2004.53.2402
 5.
de Figueiredo DG, Yang J: Decay, symmetry and existence of solutions of semilinear elliptic systems. Nonlinear Analysis: Theory, Methods & Applications 1998,33(3):211234. 10.1016/S0362546X(97)005488
 6.
Li G, Yang J: Asymptotically linear elliptic systems. Communications in Partial Differential Equations 2004,29(56):925954.
 7.
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):481504.
 8.
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):139153. 10.1007/BF01205003
 9.
Shen Y, Yao Y, Chen Z:On a class of nonlinear elliptic problem with critical potential in . Science in China Series A 2004, 34: 610624.
 10.
Cao DM:Nontrivial solution of semilinear elliptic equation with critical exponent in . Communications in Partial Differential Equations 1992,17(34):407435. 10.1080/03605309208820848
 11.
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):781791. 10.1016/j.na.2003.06.003
 12.
Kryszewski W, Szulkin A: Generalized linking theorem with an application to a semilinear Schrödinger equation. Advances in Differential Equations 1998,3(3):441472.
 13.
Willem M: Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications. Volume 24. Birkhäuser, Boston, Mass, USA; 1996:x+162.
 14.
Li G, Szulkin A: An asymptotically periodic Schrödinger equation with indefinite linear part. Communications in Contemporary Mathematics 2002,4(4):763776. 10.1142/S0219199702000853
 15.
do Ó JM, Souto MAS:On a class of nonlinear Schrödinger equations in involving critical growth. Journal of Differential Equations 2001,174(2):289311. 10.1006/jdeq.2000.3946
Acknowledgment
This work is supported by Innovation Program of Shanghai Municipal Education Commission under Grant no. 08 YZ93.
Author information
Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Zhang, G., Liu, S. Existence Result for a Class of Elliptic Systems with Indefinite Weights in . Bound Value Probl 2008, 217636 (2008). https://doi.org/10.1155/2008/217636
Received:
Accepted:
Published:
Keywords
 Weak Solution
 Elliptic Problem
 Nontrivial Solution
 Mountain Pass
 Sobolev Embedding