EXISTENCE SOLUTION FOR WEIGHTED p(x)-LAPLACIAN EQUATION

Abstract

This paper deals with the existence solution for the following type of boundary value problems:

\begin{cases} (∆ (|x|^{p(x)} |∆u|^{p(x)-2} ∆u ) = λ|u|^{q(x)−2} u, in Ω, \\ u = ∆u = 0, \; on \ ∂Ω, \end{cases}

where Ω is a smooth bounded domain in \(\mathcal{R}\)^N.  It is established for a negative λ, there exists at least one weak solution.  Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces and a variant of the Mountain Pass theorem.