The Scientific World Journal

Volume 2013 (2013), Article ID 767989, 7 pages

http://dx.doi.org/10.1155/2013/767989

Research Article

Multiple Solutions for a Class of Dirichlet Double Eigenvalue Quasilinear Elliptic Systems Involving the ()-Laplacian Operator

Armin Hadjian and Saleh Shakeri

Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, P.O. Box 678, Amol, Iran

Received 23 August 2013; Accepted 18 September 2013

Academic Editors: A. Favini and M. Spadini

Copyright © 2013 Armin Hadjian and Saleh Shakeri. 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.

Abstract.

Existence results of three weak solutions for a Dirichlet double eigenvalue quasilinear elliptic system involving the ()-Laplacian operator, under suitable assumptions, are established. Our main tool is based on a recent three-critical-point theorem obtained by Ricceri. We also give some examples to illustrate the obtained results.

1. Introduction and Preliminaries

The aim of this paper is to investigate the existence of at least three weak solutions for the following Dirichlet double eigenvalue quasilinear elliptic system:

for , where is the -Laplacian operator, is a nonempty bounded open set with a boundary of class , and are positive parameters, and for . Here, are measurable functions with respect to for every and are with respect to for a.e. , and and denote the partial derivatives of and with respect to , respectively.

Moreover, and satisfy the following additional assumptions:

for every and every ,

for a.e. ; for every and every ,

Here and in the following, we let be the Cartesian product of the Sobolev spaces for ; that is, equipped with the norm

where for ,

Put

Since for , is compactly embedded in , so that . In addition, it is known (see [1, formula (6b)]) that

for , where denotes the Gamma function defined by

and is the Lebesgue measure of the set , and equality occurs when is a ball.

Moreover, let

Simple calculations show that there is such that , where stands for the open ball in of radius centered at .

Put

for .

By a (weak) solution of system (1), we mean any such that

for all .

In the literature many papers (see, e.g., the papers [2–9] and references therein) deal with nonlinear elliptic problems. Motivated by the fact that such kind of problems is used to describe a large class of physical phenomena, many authors have studied the existence and multiplicity of solutions for (1).

The goal of this work is to establish some new criteria for system (1) to have at least three weak solutions in , by means of a very recent abstract critical point result of Ricceri [10]. We first recall the following three-critical-point theorem that follows from a combination of [11, Theorem  3.6] and [10, Theorem  1].

Lemma 1. Let be a reflexive real Banach space; let be a continuously Gâteaux differentiable and sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on , bounded on bounded subsets of ; a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact such that

Assume that there exists and , with , such that

(a₁); (a₂)for each, the functional is coercive.

Then, for each compact interval , there exists with the following property: for every and every functional with compact derivative, there exists such that, for each , the equation

has at least three solutions in whose norms are less than .

For other basic notations and definitions, we refer the reader to [12–14].

2. Main Result

In the present section we discuss the existence of multiple solutions for system (1). For any , we denote by the set

This set will be used in some of our hypotheses with appropriate choices of .

We formulate our main result as follows.

Theorem 2. Assume that there exist two positive constants and with such that

(b₁) for a.e. and all for ; (b₂)

(b₃)

uniformly with respect to .

Then, setting

for each compact interval , there exists with the following property: for every , there exists such that, for each , system (1) admits at least three weak solutions in whose norms are less than .

Proof. Our aim is to apply Lemma 1 to our problem. To this end, for each , we let the functionals be defined by

Clearly, is bounded on each bounded subset of and it is known that and are well-defined and continuously Gâteaux differentiable functionals whose derivatives at the point are the functionals and given by

for every , and is sequentially weakly lower semicontinuous (see Proposition  25.20 of [14]). Also, is a uniformly monotone operator in (for more details, see (2.2) of [15]), and since is coercive and hemicontinuous in , by applying Minty-Browder theorem (Theorem  26.A of [14]), admits a continuous inverse on .

We claim that is a compact operator. To this end, it is enough to show that is strongly continuous on . For this, for fixed , let weakly in as . Then we have that converges uniformly to on as (see [14]). Since is in for every , the derivatives of are continuous in for every , so fo , strongly as . By the Lebesgue control convergence theorem, strongly as . Thus we proved that is strongly continuous on , which implies that is a compact operator by [14, Proposition  26.2]. Hence the claim is true.

Moreover, we have

Next, put such that for ,

and . Clearly and, in particular, one has for

So

Bearing in mind that and that for , one has .

Since for each for , condition (b₁) ensures that

Hence

Moreover, owing to assumption (b₂), we have

Taking into account that for each

for (see (6)), we have that

for every , and taking into account (26) and (28), it follows that

Therefore, assumption (a₁) of Lemma 1 is satisfied.

Now, fixed , due to (b₃), there exist two constants with

such that

for all and for all . Fix . Then

for all . So, for any fixed , from (28) and (32), we have

and thus

which means that the functional is coercive. Then, also condition (a₂) of Lemma 1 holds.

In addition, since is a measurable function with respect to for every and is with respect to for a.e. , satisfying condition (G), the functional

is well defined and continuously Gâteaux differentiable on with a compact derivative, and

for all , . Thus, all the hypotheses of Lemma 1 are satisfied. Also note that the solutions of the equation

are exactly the weak solutions of (1). So, the conclusion follows from Lemma 1.

We now point out the following special case of Theorem 2 when does not depend on .

Theorem 3. Let be a -function and assume that there exist two positive constants and with such that

(b₄) for all for ; (b₅)

(b₆)

Then, setting

for each compact interval , there exists with the following property: for every , there exists such that, for each , the system

for , admits at least three weak solutions in whose norms are less than .

Proof. Set for all and for . Since , Theorem 2 ensures the conclusion.

Let and . Then we have the following existence result.

Corollary 4. Let be a continuous function and let be an -Carathéodory function. Put for each and assume that there exist two positive constants and with such that

(b₇) for all ;(b₈);(b₉)

Then, setting

for each compact interval , there exists with the following property: for every , there exists such that, for each , the problem

admits at least three weak solutions in whose norms are less than .

Now, we want to point out a simple consequence of Corollary 4 in the case when and . For simplicity, we fix and note that in this situation, and .

Corollary 5. Let be a continuous function and an -Carathéodory function. Put for each and assume that there exist two positive constants and with such that assumption (b₇) in Corollary 4 holds, and

(b₁₀);(b₁₁)

Then, setting

for each compact interval , there exists with the following property: for every , there exists such that, for each , the problem

admits at least three classical solutions in whose norms are less than .

3. Examples

First, we present an example of the application of Theorem 3.

Example 1. Let . Consider the system

where is an arbitrary function which is measurable with respect to for every and is with respect to for a.e. , satisfying

for every and . Taking into account , picking and

for each , so that and , by choosing , we have for all and

We see that

which gives that

Hence, all the assumptions of Theorem 3, with , are satisfied. So, setting

for each compact interval , there exists with the following property: for every , there exists such that, for each , system (45) has at least three weak solutions in whose norms are less than .

The next example follows directly by Corollary 5.

Example 2. Suppose for all . Then one has for all . Now, consider the following two-point boundary value problem

where is an arbitrary -Carathéodory function. Note that, in this case, , , , and . In fact, by choosing and , we have for all . Also

which shows that assumption (b₁₀) is fulfilled. Furthermore, we have . Thus, all hypotheses of Corollary 5 are satisfied. So, setting , for each compact interval , there exists with the following property: for every , there exists such that, for each , problem (52) has at least three classical solutions in whose norms are less than .

In particular, there exist two positive constants and such that, for each , the problem

admits at least three classical solutions in whose norms are less than .

References

G. Talenti, “Some inequalities of Sobolev type on two-dimensional spheres,” in General Inequalities 5, W. Walter, Ed., vol. 80 of International Series of Numerical Mathematics, pp. 401–408, Birkhäuser, Basel, Switzerland, 1987.

G. Bonanno, S. Heidarkhani, and D. O'Regan, “Multiple solutions for a class of dirichlet quasilinear elliptic systems driven by a (p, q)-Laplacian operator,” Dynamic Systems and Applications, vol. 20, no. 1, pp. 89–100, 2011.

G. Bonanno and G. Molica Bisci, “Three weak solutions for elliptic Dirichlet problems,” Journal of Mathematical Analysis and Applications, vol. 382, no. 1, pp. 1–8, 2011.

G. Bonanno, G. Molica Bisci, and V. Rdulescu, “Multiple solutions of generalized Yamabe equations on Riemannian manifolds and applications to Emden-Fowler problems,” Nonlinear Analysis: Real World Applications, vol. 12, no. 5, pp. 2656–2665, 2011.

G. Bonanno, G. Molica Bisci, and V. Rǎdulescu, “Existence of three solutions for a non-homogeneous Neumann problem through Orlicz-Sobolev spaces,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 14, pp. 4785–4795, 2011.

P. Felmer and Z. Q. Wang, “Multiplicity for symmetric indefinite functionals: application to Hamiltonian and elliptic systems,” Topological Methods in Nonlinear Analysis, vol. 12, pp. 207–226, 1998.

J. R. Graef, S. Heidarkhani, and L. Kong, “A critical points approach for the existence of multiple solutions of a Dirichlet quasilinear system,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 1268–1278, 2012.

C. Li and C.-L. Tang, “Three solutions for a class of quasilinear elliptic systems involving the (p, q)-Laplacian,” Nonlinear Analysis: Theory, Methods and Applications, vol. 69, no. 10, pp. 3322–3329, 2008.

G. Molica Bisci and V. Rdulescu, “Multiple symmetric solutions for a Neumann problem with lack of compactness,” Comptes Rendus Mathematique de l'Académie des Sciences, vol. 351, no. 1-2, pp. 37–42, 2013.

B. Ricceri, “A further refinement of a three critical points theorem,” Nonlinear Analysis: Theory, Methods and Applications, vol. 74, no. 18, pp. 7446–7454, 2011.

G. Bonanno and S. A. Marano, “On the structure of the critical set of non-differentiable functions with a weak compactness condition,” Applicable Analysis, vol. 89, no. 1, pp. 1–10, 2010.

M. Ghergu and V. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, vol. 37 of Oxford Lecture Series in Mathematics and Its Applications, Oxford University Press, 2008.

A. Kristály, V. Rădulescu, and C. Varga, Variational Principles in Mathematical Physics, Geometry, and Economics: Qualitative Analysis of Nonlinear Equations and Unilateral Problems, vol. 136 of Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, UK, 2010.

E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/B: Nonlinear Monotone Operators, Springer, New York, NY, USA, 1990.

J. Simon, “Régularite de la solution d’une équation non linéaire dans ,” in Journées d'Analyse Non Linéaire, P. Bénilan and J. Robert, Eds., vol. 665 of Lecture Notes in Mathematics, pp. 205–217, 1978.