Existence results for some anisotropic Dirichlet problems
Introduction
It is well-known that a great attention in the last years has been focused by many authors on the study of anisotropic equations on bounded Euclidean domains. See, among others, the papers [2], [3], [4], [5], [6], [9], [11], [13] as well as [14], [17], [18], [19], [22], [23], [28] and references therein.
Motivated by this large interest in the current literature, we study here the existence of weak solutions for the following anisotropic Dirichlet problem where denotes the -Laplace operator, is a bounded domain with smooth boundary ∂Ω, λ is a positive real parameter, and is a continuous function satisfying Moreover, is a continuous function such that
there exist and with for each , such that for each , where
A special case of our result reads as follows. Theorem 1.1 Let Ω be a smooth and bounded domain of the Euclidean space , , and be a continuous function for which there exist and such that for every ; there are and such that for any .
Then, there exists an open interval such that, for every , the following problem admits at least two (distinct) weak solutions in the Sobolev .
Our abstract tool for proving the main result is the following abstract theorem that we recall here in a convenient form. Theorem 1.2 Let E be a reflexive real Banach space, and let be two continuously Gâteaux differentiable functionals such that Φ is sequentially weakly lower semicontinuous and coercive. Further, assume that Ψ is sequentially weakly continuous. In addition, assume that, for each , the functional satisfies the classical compactness Palais-Smale (briefly condition. Then, for each and each the following alternative holds: either the functional has a strict global minimum which lies in , or has at least two critical points one of which lies in .
The above critical point result comes out from a joint application of the classical Pucci-Serrin theorem (see [20]) and a local minimum result due to Ricceri (see [25]). For a proof of Theorem 1.2 see, for instance, [24, Theorem 6]. We refer the interested reader to [16], [26], [27] and references therein for recent applications of the Ricceri's variational principle.
The plan of the paper is as follows. Section 2 is devoted to our abstract framework and preliminaries. Successively, in Section 3, Theorem 3.1 and some preparatory results concerning the compactness Palais-Smale condition (see Lemma 3.2, Lemma 3.3) are presented.
In the last section, Theorem 3.1 has been proved and a concrete example of an application is presented in Example 4.3.
Section snippets
Abstract framework
Here and in the sequel, we assume that verifies the previous condition and is globally log-Hölder continuous on Ω. The variable exponent Lebesgue space is defined
On we consider the Luxemburg norm The generalized Lebesgue-Sobolev space is defined by putting and it is endowed with the following norm
The main result and some technical lemmas
The aim of this section is to prove that, under natural assumptions on the nonlinear term f, weak solutions to problem below do exist. With the above notations the main result reads as follows. Theorem 3.1 Let be a continuous function such that there exist and with for each , such that for each , where there are and such that for any , and .
Then, for
Proof of Theorem 3.1
For the proof of our result, we observe that problem has a variational structure, indeed it is the Euler-Lagrange equation of the functional . Note that the functional is continuously Gâteaux differentiable in and one has for every . Thus, the critical points of are exactly the weak solutions to problem . Let and set , with λ as in the statement.
Hence, let us apply Theorem 1.2, taking
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