Existence Result for a Class of Elliptic Systems with Indefinite Weights in
© G. Zhang and S. Liu. 2008
Received: 31 October 2007
Accepted: 4 March 2008
Published: 12 March 2008
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.
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  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  obtained the existence of a nontrivial solution. As the potential and the nonlinearity are asymptotically periodic at infinity, Alves et al.  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  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:
(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;
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  proved a version of Trudinger-Moser inequality, that is,
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.
There exist and such that
There exist and such that
- (1)By assumption (H3), we have on(3.7)
- (2)Assumption (H3) implies that there exist such that(3.8)
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
From these, we have , so is weak solution of problem (P).
which is a contradiction to as
Consequently, we have a nontrivial critical point of the functional and conclude the proof of Theorem 2.2.
This work is supported by Innovation Program of Shanghai Municipal Education Commission under Grant no. 08 YZ93.
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