Uma classe de problemas de Dirichlet do tipo p(x)-laplaciano

Abstract

We study the following class of PDEs of 𝑝�(𝑥�)-laplacian type ﹛−div (︂𝜑�(𝑥�, |∇𝑢�|)∇𝑢�)︂ = 𝜆�𝑓�(𝑥�, 𝑢�), if 𝑥� ∈ Ω 𝑢�(𝑥�) = 0, if 𝑥� ∈ 𝜕�Ω, where Ω ⊂ R𝑁� is a bounded domain with a Lipschitz boundary 𝜕�Ω. We prove two existence theorems of non-trival week solutions for this problem, where one of them, insure that there exists an infinity of week solutions. Still we obtain a caracterization theorem for an eigenvalue. In both results we use the variational tecniques, in special the Mountain Pass Theorem and the Fountain’s Theorem. Yet we dedicate a whole chapter for the study about the basic results of the Lebesgue and Sobolev Spaces with Variable Exponent.