On the eigenvalue problem for a class of Kirchhoff-type equations

Abstract

We study the eigenvalue problem for the general Kirchhoff's equation −M(∫Ω|∇u(y)|p(y)dy)div(|∇u(x)|p(x)−2∇u(x))=λ|u(x)|q(x)−2u(x), for suitable M, in the context of variable exponent Lebesgue spaces. We prove that given a bounded domain Ω, there is a solution u with ∫Ω|∇u(x)|p(x)dx=r, for any r>0. A class of related problems is also treated via the same methods.