Existence of three solutions for a Navier boundary value problem involving the -biharmonic
© Huang et al.; licensee Springer. 2013
Received: 31 July 2013
Accepted: 4 September 2013
Published: 7 November 2013
In this paper, we study -biharmonic systems with Navier boundary condition on a bounded domain and obtain three solutions under appropriate hypotheses. The technical approach is mainly based on the general three critical points theorem obtained by Ricceri.
1 Introduction and main results
is the critical exponent just as in many papers. Obviously, , for all .
In what follows, E denotes the Cartesian product of two Sobolev spaces and , i.e., , and X denotes the Sobolev space .
In recent years, the study of differential equations and variational problems with -growth conditions has been an interesting topic resulting from nonlinear electrorheological fluids (see ) and elastic mechanics (see ).
Some authors considered elliptic systems (see [3–16]) which have been used in a wide range of applications. Existence and multiplicity results for elliptic systems involving variational structure have been extensively investigated.
where p and q are real numbers larger than 1.
In , applying the fibering method established by Pohozaev, Bozhkova and Mitidieri, the authors studied the existence of multiple solutions for quasilinear system involving a pair of -Laplacian operators.
where , and F satisfies suitable assumptions.
In , Afrouzi and Heidarkhani studied the existence of three solutions for a class of Dirichlet quasilinear elliptic systems involving the -Laplacian. In , Jing-Jing Liu and Xia-Yang Shi proved the existence of multiple solutions for a quasilinear system involving a pair of -Laplacian operators. In , Bin Ge and Ji-Hong Shen obtained multiple solutions for a class of differential inclusion systems involving the -Laplacian.
where , , and F, G satisfy suitable assumptions.
The main result of this paper is the following theorem.
Theorem 1.1 Suppose that there exist two positive constants C, d and two functions with , such that
(j1) for a.e. and all ;
(j3) for a.e. and all ;
where and , for all , there exists such that, for each , problem (P) has at least three weak solutions whose norms in are less than r.
The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge about the Lebesgue and Sobolev spaces with variable exponents, and present Ricceri’s three-critical-points theorem. In Section 3, we prove the main result.
We denote by the closure of in .
Proposition 2.1 (see )
The spaces , and are separable and reflexive Banach spaces;
If and for any , then the imbedding from to is compact and continuous;
There is a constant such that .
By (iii) of Proposition 2.1, we know that and are equivalent norms on . We use to replace in the following discussion.
Proposition 2.2 (see )
For , ;
If , then ;
If , then ;
Similar to Proposition 2.2, we obtain the following.
For , ;
If , then ;
If , then ;
Proposition 2.4 (see )
is a continuous, bounded and strictly monotone operator;
L is a mapping of type , i.e., if in X and , then in X;
is a homeomorphism.
Proposition 2.5 (see )
has at least three solutions in X whose norms are less than r.
Proposition 2.6 (see )
3 Proof of the main result
Then is a functional and the critical points of it are weak solutions of problem (P).
In view of Proposition 2.4 (or  for details), certainly, Φ is a continuously Gâteaux differentiable and sequentially weakly lower semi-continuous functional whose Gâteaux derivative admits a continuous inverse on . Moreover, J and Ψ are continuously Gâteaux differentiable functionals whose Gâteaux derivatives are compact. Obviously, Φ is bounded on each bounded subset of X under our assumptions.
Then assumption (2.1) of Proposition 2.5 is fulfilled.
Then there exist and such that and (2.3) is satisfied.
- (i)For , by (3.2) we have
- (ii)For , from (3.1) we get
Put , moreover, , fulfil the assumption of Proposition 2.6. So, applying Proposition 2.6, we can easily get that (2.2) is fulfilled.
Thus, Φ, J and Ψ fulfil all the assumptions of Proposition 2.5, and our conclusion follows from Proposition 2.5. □
for each and as in the proof of Theorem 1.1 (namely, ).
The authors would like to thank the editors and the referees for their valuable suggestions to improve the quality of this paper. This work was supported by the Scientific Research Project of Guangxi Education Department (No. 201204LX672).
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