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 LeraySchauder.
1. Introduction
In this work, we consider the boundary value problem
where is a nonempty bounded open set with smooth boundary is the socalled Laplacian operator, and (CAR): is a Caratheodory function which satisfies the growth condition
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
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
We define a norm, the socalled Luxemburg norm,on this space by the formula
and becomes a Banach space.
The variable exponent Sobolev space is
and we define on this space the norm
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
(i)
(ii)
(iii)
(iv)
Proposition 1.3 (see [3]).
Assume that is bounded and smooth. Denote by
(i)Let . If
then is compactly imbedded in
(ii)(Poincaré inequality, see [1, Theorem ]). If , then there is a constant such that
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
Proof.
By Proposition 1.2(iv), we have
By the mean value theorem, there exists such that
and we have
Similarly
Remark 1.5.
If , then
For simplicity of notation, we write
In [4], a topological method, based on the fundamental properties of the LeraySchauder 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 LeraySchauder 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
has a solution in .
Indeed, if satisfies (1.16) then, for all , one has
which rewrites as
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
Thus, proving that problem (P) has a weak solution in reduces to proving that the compact operator
has a fixed point.
Theorem 1.6 (Alternative of LeraySchauder, [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 LeraySchauder.
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
where is given by . Notice that, according to a wellknown 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
where is the constant appearing in condition (CAR). Let be a constant such that
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
Therefore, we have
Substituting in the above inequality, we obtain
which, taking into account (2.3) and gives
a contradiction. Theorem 2.1 applies.
Acknowledgment
The authors would like to thank the referees for their valuable and useful comments.
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