Eigenvalue problems for the p-Laplacian with indefinite weights

Abstract

We consider the eigenvalue problem $-Delta_p u=lambda V(x) |u|^{p-2} u, uin W_0^{1,p} (Omega)$ where $p>1$, $Delta_p$ is the p-Laplacian operator, $lambda >0$, $Omega$ is a bounded domain in $mathbb{R}^N$ and $V$ is a given function in $L^s (Omega)$ ($s$ 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.

Keywords

• Nonlinear eigenvalue problem
• p-Laplacian
• indefinite weight.