| Summary: | 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.
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