Existence and Localization Results for -Laplacian via Topological Methods
© B. Cekic and R. A. Mashiyev. 2010
Received: 23 February 2010
Accepted: 20 June 2010
Published: 1 July 2010
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.
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.
For let , for a.e. .
and becomes a Banach space.
for all The space is the closure of in .
If , then the spaces , , and are separable and reflexive Banach spaces.
If and then we have
Proposition 1.3 (see ).
Assume that is bounded and smooth. Denote by
then is compactly imbedded in
Consequently, and are equivalent norms on . In what follows, , with , will be considered as endowed with the norm .
In , 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 . 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 , solutions are found in Sobolev spaces.
Let us recall some results borrowed from Dinca  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.
has a solution in .
and tells us that is a weak solution in to problem (P)
has a fixed point.
Theorem 1.6 (Alternative of Leray-Schauder, ).
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).
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
where is given by . Notice that, according to a well-known regularity result , 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.
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)
Then the Dirichlet problem (P) has at least a solution in with
a contradiction. Theorem 2.1 applies.
The authors would like to thank the referees for their valuable and useful comments.
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