Open Access

Multiple Solutions for a Class of -Laplacian Systems

Journal of Inequalities and Applications20092009:191649

Received: 19 November 2008

Accepted: 11 February 2009

Published: 18 February 2009


We study the multiplicity of solutions for a class of Hamiltonian systems with the -Laplacian. Under suitable assumptions, we obtain a sequence of solutions associated with a sequence of positive energies going toward infinity.


Banach SpaceWeak SolutionOpen SubsetBounded DomainVariational Problem

1. Introduction and Main Results

Since the space and were thoroughly studied by Kováčik and Rákosník [1], variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In [2], Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids.

In recent years, the differential equations and variational problems with -growth conditions have been studied extensively; see for example [36]. In [7], De Figueiredo and Ding discussed the multiple solutions for a kind of elliptic systems on a smooth bounded domain. Motivated by their work, we will consider the following sort of -Laplacian systems with "concave and convex nonlinearity":
where is a bounded domain, is continuous on and satisfies , and is a function. In this paper, we are mainly interested in the class of Hamiltonians such that
where Here we denote

and denote by the fact that Throughout this paper, satisfies the following conditions:

(H1) Writing

(H2) there exist such that

where is positive constant;

(H3) there exist with , and such that


As [8,Lemma 1.1], from assumption (H3), there exist such that
for any We can also get that there exists such that

for any In this paper, we will prove the following result.

Theorem 1.1.

Assume that hypotheses (H1)–(H3) are fulfilled. If is even in , then problem (1.1) has a sequence of solutions such that


2. Preliminaries

First we recall some basic properties of variable exponent spaces and variable exponent Sobolev spaces where is a domain. For a deeper treatment on these spaces, we refer to [1, 911].

Let be the set of all Lebesgue measurable functions and

The variable exponent space is the class of all functions such that Under the assumption that is a Banach space equipped with the norm (2.1).

The variable exponent Sobolev space is the class of all functions such that and it can be equipped with the norm
For if we define

then and are equivalent norms on

By we denote the subspace of which is the closure of with respect to the norm (2.2) and denote the dual space of by We know that if is a bounded domain, and are equivalent norms on

Under the condition is a separable and reflexive Banach space, then there exist and such that
In the following, we will denote that where
For any define the norm For any set and

denote the complement of in by

3. The Proof of Theorem 1.1

Definition 3.1.

We say that is a weak solution of problem (1.1), that is,

In this section, we denote that for any , and is positive constant, for any

Lemma 3.2.

Any sequence , that is, and as is bounded.


Let be sufficiently small such that

Let be such that as We get
As by the Young inequality, we can get that for any
Let be sufficiently small such that
Note that by the Young inequality, for any we get
Let be sufficiently small such that and then we get
Note that
and for being large enough, we have
It is easy to know that if and
thus we get

then are bounded. Similarly, if or we can also get that are bounded. It is immediate to get that is bounded in

Lemma 3.3.

Any sequence contains a convergent subsequence.


Let be a sequence. By Lemma 3.2, we obtain that is bounded in As is reflexive, passing to a subsequence, still denoted by we may assume that there exists such that weakly in Then we can get weakly in . Note that
It is easy to get that
and in in as Then
as By condition (H2), we obtain
It is immediate to get that are bounded and then we get
as Similar to [3, 4], we divide into two parts:
On we have
then On we have

Thus we get Then in as Similarly, in

Lemma 3.4.

There exists such that for all with


For any we have
In the following, we will consider
  1. (i)
    If We have
  1. (ii)
    If Note that For any there exists which is an open subset of such that
then is an open covering of As is compact, we can pick a finite subcovering for Thus there exists a sequence of open set such that and
for Denote that then we have

where As is a finite dimensional space, we have for

We denote by the maximum of polynomial on for Then there exists such that

for and where

Let If we get or

(i)If It is easy to verify that there exists at least such that thus
(ii)If We obtain

Now we get the result.

Lemma 3.5.

There exist and such that for any with


For By condition (H2), there exists such that
Let we get
Denote that
By [5, Lemma 3.3], we get that as then

when is sufficiently large and It is easy to get that as

Lemma 3.6.

I is bounded from above on any bounded set of


For We get
By conditions (H2) and (H3), we know that if and if Then

and it is easy to get the result.


By Lemmas 3.2–3.6 above, and [7, Proposition 2.1 and Remark 2.1], we know that the functional has a sequence of critical values as Now we complete the proof.



This work is supported by Science Research Foundation in Harbin Institute of Technology (HITC200702) and The Natural Science Foundation of Heilongjiang Province (A2007-04).

Authors’ Affiliations

Department of Mathematics, Harbin Institute of Technology, Harbin, China


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© Y. Fu and X. Zhang. 2009

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