Gradient and Parameter Dependent Dirichlet (<i>p</i>(<i>x</i>),<i>q</i>(<i>x</i>))-Laplace Type Problem

Abstract

We analyze a Dirichlet (p(x),μq(x))-Laplace problem. For a gradient dependent nonlinearity of Carathéodory type, we discuss the existence, uniqueness and asymptotic behavior of weak solutions, as the parameter μ varies on the non-negative real axis. The results are obtained by applying the properties of pseudomonotone operators, jointly with certain a priori estimates.

Keywords

• Lebesgue and Sobolev spaces with variable exponents
• parametric problems
• gradient dependent term
• Nemitsky map
• pseudomonotone operator