On eigenvalue problems of $p$-Laplacian with Neumann boundary conditions

Abstract:

We study the nonlinear eigenvalue problem \[ - {\Delta _p}u = \lambda m(x)|u{|^{p - 2}}u \quad {\text {in}} \Omega ,\quad \frac {{\partial u}} {{\partial n}} = 0 \quad {\text {on}} \partial \Omega , \;{\text {where}} p > 1,\lambda \in {\mathbf {R}}.\] For $\int _\Omega {m(x) < 0}$, we prove that the first positive eigenvalue ${\lambda _1}$ exists and is simple and unique, in the sense that it is the only eigenvalue with a positive eigenfunction. In the case $\int _\Omega {m(x) = 0}$, we prove that ${\lambda _0} = 0$ is the only eigenvalue with a positive eigenfunction.