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

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.

1. Introduction

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

(1.1)

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:

(1.2)

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

(1.3)

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:

(1.4)

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

(1.5)

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

(1.6)

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

(1.7)

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]

(2.1)

and denote the product space endowed with the inner product:

(2.2)

If we define

(2.3)

It is easy to check that since

(2.4)

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

(2.5)

Now, we define the functional

(2.6)

where

(2.7)

Let and let we define

(2.8)

Here, we assume the following condition:

(H1)

(H2) uniformly with respect to

(H3) there exist and such that

(2.9)

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

(2.10)

Then, there exist and a sequence such that

(2.11)

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,

(i)if , we have

(2.12)

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

(2.13)

Deffinition.

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

(2.14)

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

(3.1)

Then, we obtain

(3.2)

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

(3.3)

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

(3.4)

and thus on we have

(3.5)

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

(3.6)

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

(3.9)

Because it follows that for

(3.10)

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

(3.11)

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

(3.12)

Claim 3.3.

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

(3.13)

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

(3.14)

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

(3.15)

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

(3.16)

that is,

(3.17)

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

(3.18)

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

(3.19)

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

(3.20)

Hence, we have

(3.21)

for some positive constant So we have

(3.22)

Using a similar argument, we obtain

(3.23)

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

(3.24)

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

(3.25)

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

(3.26)

Similarly, we can obtain

(3.27)

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

(3.28)

where Choosing suitable and we have

(3.29)

Then, we obtain

(3.30)

Since in as this will lead to

(3.31)

Similarly, we have

(3.32)

Using assumption (H3), we obtain

(3.33)

This together with we have

(3.34)

Thus, we see that

(3.35)

which is a contradiction to as

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