The first eigencurve for a Neumann boundary problem involving p-Laplacian with essentially bounded weights

Abstract

This article is intended to prove the existence and uniqueness of the first eigencurve, for a homogeneous Neumann problem with singular weights associated with the equation $-\Delta_{p} u=\alpha m_{1}|u|^{p-2}u+\beta m_{2}|u|^{p-2}u$ in a bounded domain $\Omega \subset \mathbb{R}^{N}$. We then establish many properties of this eigencurve, particularly the continuity, variational characterization, asymptotic behavior, concavity and the differentiability.