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Existence Result for a Class of Elliptic Systems with Indefinite Weights in
Boundary Value Problems volume 2008, Article number: 217636 (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.
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  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  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  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 , de Figueiredo et al.  discussed the solvability of problems of the type
where is some bounded domain in Shen et al.  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  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
Consider the Hilbert space 
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
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
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,
(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.
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
There exist and such that
By assumption (H3), we have on(3.7)
because for any
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
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
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).
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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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
- Weak Solution
- Elliptic Problem
- Nontrivial Solution
- Mountain Pass
- Sobolev Embedding