- Research Article
- Open Access

- 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

## 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,

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:

(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;

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.

Proof.

Lemma 3.2.

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

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