Open Access

Existence and Localization Results for -Laplacian via Topological Methods

Fixed Point Theory and Applications20102010:120646

DOI: 10.1155/2010/120646

Received: 23 February 2010

Accepted: 20 June 2010

Published: 1 July 2010

Abstract

We show the existence of a week solution in to a Dirichlet problem for in , and its localization. This approach is based on the nonlinear alternative of Leray-Schauder.

1. Introduction

In this work, we consider the boundary value problem
(P)
where is a nonempty bounded open set with smooth boundary    is the so-called -Laplacian operator, and (CAR): is a Caratheodory function which satisfies the growth condition
(1.1)

with ,   for a.e. , and , for a.e. .

We recall in what follows some definitions and basic properties of variable exponent Lebesgue and Sobolev spaces , , and . In that context, we refer to [1, 2] for the fundamental properties of these spaces.

Set
(1.2)

For let , for a.e. .

Let us define by the set of all measurable real functions defined on . For any we define the variable exponent Lebesgue space by
(1.3)
We define a norm, the so-called Luxemburg norm, on this space by the formula
(1.4)

and becomes a Banach space.

The variable exponent Sobolev space is
(1.5)
and we define on this space the norm
(1.6)

for all The space is the closure of in .

Proposition 1.1 (see [1, 2]).

If , then the spaces , ,  and   are separable and reflexive Banach spaces.

Proposition 1.2 (see [1, 2]).

If   and   then we have

(i)

(ii)

(iii)

(iv)

Proposition 1.3 (see [3]).

Assume that is bounded and smooth. Denote by

(i)Let . If
(1.7)

then is compactly imbedded in

(ii)(Poincaré inequality, see [1, Theorem ]). If , then there is a constant such that
(1.8)

Consequently, and   are equivalent norms on . In what follows, , with , will be considered as endowed with the norm .

Lemma 1.4.

Assume that and If , then we have
(1.9)

Proof.

By Proposition 1.2( iv), we have
(1.10)
By the mean value theorem, there exists such that
(1.11)
and we have
(1.12)
Similarly
(1.13)

Remark 1.5.

If , then
(1.14)
For simplicity of notation, we write
(1.15)

In [4], a topological method, based on the fundamental properties of the Leray-Schauder degree, is used in proving the existence of a week solution in to the Dirichlet problem (P) that is an adaptation of that used by Dinca et al. for Dirichlet problems with classical -Laplacian [5]. In this work, we use the nonlinear alternative of Leray-Schauder and give the existence of a solution and its localization. This method is used for finding solutions in Hölder spaces, while in [6], solutions are found in Sobolev spaces.

Let us recall some results borrowed from Dinca [4] about -Laplacian and Nemytskii operator . Firstly, since for all , is compactly embedded in . Denote by the compact injection of in and by , for all , its adjoint.

Since the Caratheodory function satisfies (CAR), the Nemytskii operator generated by , , is well defined from into , continuous, and bounded ([3, Proposition ]). In order to prove that problem (P) has a weak solution in it is sufficient to prove that the equation
(1.16)

has a solution in .

Indeed, if satisfies (1.16) then, for all , one has
(1.17)
which rewrites as
(1.18)

and tells us that is a weak solution in to problem (P)

Since is a homeomorphism of onto (1.16) may be equivalently written as
(1.19)
Thus, proving that problem (P) has a weak solution in reduces to proving that the compact operator
(1.20)

has a fixed point.

Theorem 1.6 (Alternative of Leray-Schauder, [7]).

Let denote the closed ball in a Banach space and let be a compact operator. Then either

(i)the equation has a solution in for or

(ii)there exists an element with satisfying for some

2. Main Results

In this work, we present new existence and localization results for -solutions to problem (P), under (CAR) condition on Our approach is based on regularity results for the solutions of Dirichlet problems and again on the nonlinear alternative of Leray-Schauder.

We start with an existence and localization principle for problem (P).

Theorem 2.1.

Assume that there is a constant independent of , with for any solution to

and for each . Then the Dirichlet problem (P) has at least one solution with

Proof.

By [3, Theorem ], is a homeomorphism of onto We will apply Theorem 2.1 to and to operator  
(2.1)

where is given by . Notice that, according to a well-known regularity result [4], the operator from to is well defined, continuous, and order preserving. Consequently, is a compact operator. On the other hand, it is clear that the fixed points of are the solutions of problem (P). Now the conclusion follows from Theorem 1.6 since condition (ii) is excluded by hypothesis.

Theorem 2.2 immediately yields the following existence and localization result.

Theorem 2.2.

Let , be a smooth bounded domain and let be such that for all . Assume that is a Caratheodory function which satisfies the growth condition (CAR)

Suppose, in addition, that
(2.2)
where is the constant appearing in condition (CAR). Let be a constant such that
(2.3)

Then the Dirichlet problem (P) has at least a solution in with

Proof.

Let be a solution of problem ( ) with , corresponding to some . Then by Propositions 1.2, 1.3, and Lemma 1.4, we obtain
(2.4)
Therefore, we have
(2.5)
Substituting in the above inequality, we obtain
(2.6)
which, taking into account (2.3) and gives
(2.7)

a contradiction. Theorem 2.1 applies.

Declarations

Acknowledgment

The authors would like to thank the referees for their valuable and useful comments.

Authors’ Affiliations

(1)
Department of Mathematics, Dicle University

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Copyright

© B. Cekic and R. A. Mashiyev. 2010

This article is published under license to BioMed Central Ltd. 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.