Multiple Solutions for a Class of -Laplacian Systems
© Y. Fu and X. Zhang. 2009
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.
1. Introduction and Main Results
Since the space and were thoroughly studied by Kováčik and Rákosník , variable exponent Sobolev spaces have been used in the last decades to model various phenomena. In , Růžička presented the mathematical theory for the application of variable exponent spaces in electro-rheological fluids.
and denote by the fact that Throughout this paper, satisfies the following conditions:
where is positive constant;
for any In this paper, we will prove the following result.
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).
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
denote the complement of in by
3. The Proof of Theorem 1.1
In this section, we denote that for any , and is positive constant, for any
Any sequence , that is, and as is bounded.
Let be sufficiently small such that
then are bounded. Similarly, if or we can also get that are bounded. It is immediate to get that is bounded in
Any sequence contains a convergent subsequence.
Thus we get Then in as Similarly, in
There exists such that for all with
- (i)If We have(321)
- (ii)If Note that For any there exists which is an open subset of such that(322)
where As is a finite dimensional space, we have for
for and where
Let If we get or
Now we get the result.
There exist and such that for any with
when is sufficiently large and It is easy to get that as
I is bounded from above on any bounded set of
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).
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