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Existence of three solutions for a Navier boundary value problem involving the -biharmonic
Boundary Value Problems volume 2013, Article number: 228 (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
In this paper, we consider the Navier boundary value problem involving the -biharmonic systems
where , () is a nonempty bounded open set with a boundary ∂ Ω of class , are functions such that , are measurable in Ω for all and is in for a.e. , denotes the partial derivative of F with respect to i, , so does . And , , . Moreover,
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.
In , Boccardo and Figueiredo studied the following boundary value problem involving the -Laplacian:
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.
In , Chun Li and Chun-Lei Tang ensured the existence of three solutions for the problem
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.
Recently, Lin Li and Chun-Lei Tang (see ) considered the Navier boundary value problem involving the -biharmonic systems
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 ;
(j2) and , such that
(j3) for a.e. and all ;
(j4) for a.e. . Then there exist an open interval and a positive real number r with the following property: for each and each function , measurable in Ω, in and satisfying
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.
Assume that Ω is a bounded domain of () with a smooth boundary ∂ Ω. Let
For , set
For , define
with the norm
endowed with the norm
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 )
Set for , we obtain
For , ;
If , then ;
If , then ;
In this paper, the space E is endowed with the following equivalent norm:
Similar to Proposition 2.2, we obtain the following.
Proposition 2.3 Let for , we obtain:
For , ;
If , then ;
If , then ;
Let , and denote , 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 )
Let X be a separable and reflexive real Banach space; ; let be a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on ; is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. In addition, Φ is bounded on each bounded subset of X. Assume that
for all , and that there exists such that
Then there exist a nonempty open set and a positive real number r with the following property: for every and every functional with a compact derivative, there exists such that, for each , the equation
has at least three solutions in X whose norms are less than r.
Proposition 2.6 (see )
Let X be a nonempty set, and let Φ, J be two real functionals on X. Assume that there are and such that
Then, for each ρ satisfying
3 Proof of the main result
Definition 3.1 A weak solution of problem (P) is any such that
for all . We define the corresponding energy functional of problem (P) as
Then is a functional and the critical points of it are weak solutions of problem (P).
Proof of Theorem 1.1 Let Φ, J, Ψ as above. So, for each , one has
Therefore, the weak solutions of problem (P) are exactly the solutions of the equation
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.
By Proposition 2.3, set just as before, then for ,
Actually, for , set , then we can obtain
Hence we have
So, there exists a constant such that
holds for any .
holds for any , where constants , . Here, by using conditions (j3) and (ii) of Proposition 2.1, combining the two inequalities above, we can obtain
Due to , , we get
Then assumption (2.1) of Proposition 2.5 is fulfilled.
Next, we derive that assumption (2.2) is also fulfilled. It is easy to verify the conditions of Proposition 2.6. Let , we can easily have
Then there exist and such that and (2.3) is satisfied.
There is a point since it is a nonempty bounded open set. Let , put
where is the open ball in of radius r centered at x,
Let , then by (j1) we can derive that
From (j2), , such that
By (j3), there are nine positive real numbers () according to , larger or smaller than η and 1. For example, when , some
Set , then
Hence, fix γ such that . And for , by the Sobolev embedding theorem ( is continuous), there exist suitable positive constants and such that
Since , , we have
We choose as above such that . Fix such that . Then we divide the proof into two cases.
For , by (3.2) we have
By (3.3), we obtain
For , from (3.1) we get
From (3.3), we have
For any , we can obtain , i.e.,
Then we can have
This inequality implies
Therefore we have
So, we can get that
Hence we can find , and satisfying (2.3). Also, we can find ρ satisfying
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. □
Remark Applying Theorem 2.1 in  to the proof of Theorem 1.1, an upper bound of the interval of parameters λ, for which (P) has at least three weak solutions, is obtained. To be precise, in the conclusion of Theorem 1.1, one has
for each and as in the proof of Theorem 1.1 (namely, ).
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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).
The authors declare that they have no competing interests.
This paper is the result of joint work of all authors who contributed equally to the final version of this paper. All authors read and approved the final manuscript.