Eigenvalue Problems for the p-Laplacian with Indefinite Weights

Abstract

We consider the eigenvalue problem -∆pu = λV(x) |u|p-2 u, u ∈ W1,p0 (Ω) where p > 1, ∆p is the p-Laplacian operator, λ > 0, Ω is a bounded domain in ℝN and V is a given function in Ls (Ω) (s depending on p and N). The weight function V may change sign and has nontrivial positive part. We prove that the least positive eigenvalue is simple, isolated in the spectrum and it is the unique eigenvalue associated to a nonnegative eigenfunction. Furthermore, we prove the strict monotonicity of the least positive eigenvalue with respect to the domain and the weight.