Steklov problem with an indefinite weight for the p-Laplacian

Abstract

Let $Omegasubsetmathbb{R}^{N}$, with $Ngeq2$, be a Lipschitz domain and let 1 lees than p less than $infty$. We consider the eigenvalue problem $Delta_{p}u=0$ in $Omega$ and $| abla u|^{p-2}frac{partial u}{partial u}=lambda m|u|^{p-2}u$ on $partialOmega$, where $lambda$ is the eigenvalue and $uin W^{1,p}(Omega)$ is an associated eigenfunction. The weight $m$ is assumed to lie in an appropriate Lebesgue space and may change sign. We sketch how a sequence of eigenvalues may be obtained using infinite dimensional Ljusternik-Schnirelman theory and we investigate some of the nodal properties of eigenfunctions associated to the first and second eigenvalues. Amongst other results we find that if $m^{+} otequiv 0$ and $int_{partialOmega} m,dsigma<0$ then the first positive eigenvalue is the only eigenvalue associated to an eigenfunction of definite sign and any eigenfunction associated to the second positive eigenvalue has exactly two nodal domains.

Keywords

• Nonlinear eigenvalue problem
• Steklov problem
• p-Laplacian
• nonlinear boundary condition
• indefinite weight.