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Miao, Qing

doi:https://doi.org/10.1155/2016/3050417

Discrete Dynamics in Nature and Society

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Abstract Introduction Preliminaries Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2016 | Article ID 3050417 | https://doi.org/10.1155/2016/3050417

Multiple Solutions for Nonlinear Navier Boundary Systems Involving -Biharmonic Problem

Qing Miao¹

Academic Editor: Pasquale Candito

Received01 Dec 2015

Revised25 Jan 2016

Accepted08 Feb 2016

Published02 Mar 2016

Abstract

We improve some results on the existence and multiplicity of solutions for the -biharmonic system. Our main results are new. Our approach is based on general variational principle and the theory of the variable exponent Sobolev spaces.

1. Introduction

In this paper, we consider the existence of solutions for the following system: for , where is a bounded domain with smooth boundary . is a positive parameter and is a function such that the mapping is in in for all , denotes the partial derivative of with respect to , and is continuous in , for . with .

In recent years, many authors considered the existence and multiplicity of solutions for some fourth order problems [1–10]. In [4], based on critical point theory, the existence of infinitely many solutions has been established for a class of nonlinear elliptic equations involving the -biharmonic operator and under Navier boundary value conditions. The -Laplacian operator is more complicated nonlinearities than -Laplacian; it is inhomogeneous and usually it does not have the so-called first eigenvalue, since the infimum of its principle eigenvalue is zero. In [11], based on variational methods, the authors established the existence of an unbounded sequence of weak solutions for a class of differential equations with -Laplacian. In [12], when the nonlinearity has the subcritical growth and via variational methods [13], the authors obtained the existence of at least one, two, or three weak solutions for a class of differential equations with -Laplacian whenever the parameter belongs to a precise positive interval. Recently, the -biharmonic problems have attracted more and more attention; we refer the reader to [11, 14–21]. In [16], El Amrouss and Ourraoui studied the -biharmonic equation with Navier and Neumann boundary condition; the technical approach is based on Ricceri’s variational principle and local mountain pass theorem, without Palais-Smale condition. In [20], the authors established the existence of at least three solutions for elliptic systems involving the -biharmonic operator. In [15], Allaoui et al. considered the existence of infinitely many solutions for the -biharmonic problem by a general Ricceri’s variational principle. However, there are rare results on -biharmonic problem.

Inspired by the aforementioned papers, our objective is to prove the existence and multiplicity solutions for problem (1); we study problem (1) by using the results as follows.

Theorem A (see [13, 22]). Let be a reflexive real Banach space; is 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 such that Assume that there exist and , with , such that(i);(ii)for each , the functional is coercive.Then, for each compact interval , there exists with the following property: for every , the equation has at least three solutions in whose norms are less than .

Theorem B (see [23]). Let be a reflexive real Banach space; are two Gâteaux differentiable functionals such that is sequentially weakly lower semicontinuous and coercive and is sequentially weakly upper semicontinuous. For every , let one put Then, one has the following:(a)For every and every , the restriction of the functional to admits a global minimum, which is a critical point (local minimum) of in .(b)If , then, for each , the following alternative holds: either(b1) possesses a global minimum, or(b2)there is a sequence of critical points (local minima) of such that .(c)If , then, for each , the following alternative holds: either(c1)there is a global minimum of which is a local minimum of , or(c2)there is a sequence of pairwise distinct critical points local minima of which weakly converges to a global minimum of .

This paper is organized as follows. In Section 2, we recall some basic facts about the variable exponent Lebesgue and Sobolev spaces, some important properties of the -biharmonic operator. In Section 3, we establish the main results.

2. Preliminaries

In order to deal with the -biharmonic problem, we need some theories on spaces , and introduce some notations used in the following.

Denote

We introduce a norm on : Then becomes a Banach space; we call it a generalized Lebesgue space.

Proposition 1 (see [24]). The conjugate space of is , where . For any and , one has the following Hölder-type inequality: The variable exponent Sobolev space is defined bywhere is the multi-index and is the order, is a positive integer, and it can be equipped with the norm From [24], we know that spaces and are separable, reflexive, and uniform convex Banach spaces.
We denote by the closure of in .
Let endow with the norm where

Remark 2. According to [25], the norm is equivalent to the norm in the space . Consequently, the norms , , and are equivalent.

Proposition 3 (see [24]). Put ; then(1);(2);(3);(4).

Proposition 4 (see [20, 26]). The embedding is compact whenever . So there is a constant such that

3. Main Results

Definition 5. One says that is a weak solution to the system (1) if and for every .
Let , . For , one denotes the set Define the function by for all , where Then the operator , where is the dual space of , is defined by for .

Proposition 6. is continuous, coercive, and strictly monotone. admits a continuous inverse on .

Proof. Sinceand , then is coercive.
Using the elementary inequalities We deduce that which means that is strictly monotone. The inverse operator of exists and the continuity of can be proved essentially by the same way as the latter part of the proof of [16, Proposition ]; we omit the details.

From Proposition 6, we see that . Since is compactly embedded in , we can see that are sequentially weakly lower semicontinuous.

The functional is Gateaux differentiable functional and for . is sequentially weakly upper semicontinuous. Furthermore, is a compact operator. Indeed, it is enough to show that is strongly continuous on . For this, for fixed , let weakly in as . Then we have converges uniformly to on as [27]. Since is in for every , so for , strongly as , from which follows strongly as . Thus we have that is strongly continuous on , which implies that is a compact operator by [27, Proposition ].

Theorem 7. Assume the following: (A1) for .(A2)There exist and positive constants with for , such that for a.e. , , where , , for .(A3)There exist , , , and with and such that where for .Then, setting for each compact interval , there exists a positive real number with the following property: for every , problem (1) admits at least three weak solutions whose norms are less than .

Proof. To apply Theorem A to our problem, the functionals satisfy the conditions of Theorem A. Now, we show that the hypotheses of Theorem A are fulfilled.
Now we set ; from (A1), we have . Let , , and take Let , and . Clearly, , and we have On the other way, when , we have So, by Proposition 3, we have We deduce that For , we have for .
From (12) we have ; we obtain for all , It follows that, for every , Since therefore, from (A3), we have and the assumption (i) of Theorem A is satisfied.
From Proposition 3, we know that if , then let , such that , and then If , then From (A2), (12), (35), and (36), we have noting that ; therefore for , we see that in particular, for every . Then the assumption (ii) of Theorem A holds.
Then all the assumptions of Theorem A are fulfilled. By Theorem A, we know that there exist an open interval and a positive constant such that, for any , problem (1) has at least three weak solutions whose norms are less than .

Remark 8. Graef et al. [5] studied the problem and established the existence of at least three solutions in the particular case when .

Theorem 9. Assume the following: (A4), for each .(A5)There exist , such that, if one puts one has where , Then, for every problem (1) admits an unbounded sequence of weak solutions.

Proof. To apply Theorem B to our problem, the functionals satisfy the conditions of Theorem B. Now, let us verify that . Let be a real sequence such that as and Put for all , So, by Proposition 3, we have Hence for large enough (),From (12) we have ; we obtain for all , Thus . Then we have Note that , ; then Therefore, from (A5), we have It is clear that .
For the fixed , the other step is to show that the functional has no global minimum. Arguing as in [15], since , we can consider positive real sequences and such that as and Let be a sequence in defined by for . For any fixed , it is to see that , and Then By (A1), we have and combining (51), (54), and (55), we obtain for large enough, so Hence, our claim is proved. Since all assumptions of Theorem B case (b) are satisfied, the functional admits an unbounded sequence of critical points. This completes the proof of Theorem 9.

Theorem 10. Assume that (A1), (A4) hold and consider the following: (A6)There exist , such that, if one puts one has where , Then, for every problem (1) admits a sequence of weak solutions which converges to .

Proof. From condition (A1), we have , .
Let be a real sequence such that as and Put for all . Therefore, from (A6), we have It is clear that .
For the fixed , the other step is to show that the functional has not a local minimum at zero. Arguing as in [15], since , we can consider positive real sequences and such that as and Let be a sequence in defined by (52):Combining (55), (64), and (65), for large enough, we have The alternative of Theorem B case (c) ensures the existence of sequence of pairwise distinct critical points (local minima) of which weakly converges to 0. This completes the proof of Theorem 10.

Example 11. Let , with being three functions defined on by , , and , and consider the increasing sequence of positive real numbers given by Define the function by where denotes the open unit ball with center at . It is easy to verify that is nonnegative function such that is continuous in for all . is in for every . for all , for every : The restriction of on attains its maximum in and .
Hence Therefore Moreover, by choosing , for every , we have then and so Then, Hence, from Theorem 9, for each , the problem admits an unbounded sequence of weak solutions.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research is partly supported by the National Natural Science Foundation of China (11461083, 11561076), the Applied Basic Research Projects of Yunnan Province (2013FD031), and the Project of Science Research Fund of Yunnan Education Department (2015Y222).

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Copyright © 2016 Qing Miao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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