Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 87, pp. 1-9.

Title: Steklov problem with an indefinite weight for the p-Laplacian

Author: Olaf Torne (Univ. Libre de Bruxelles, Belgium)

Abstract: 
 Let $\Omega\subset\mathbb{R}^{N}$, with $N\geq2$, be a Lipschitz
 domain and let $1<p<\infty$.
 We consider the eigenvalue problem $\Delta_{p}u=0$ in $\Omega$ and
 $|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}=\lambda m|u|^{p-2}u$
 on $\partial\Omega$, where $\lambda$ is the eigenvalue and
 $u\in W^{1,p}(\Omega)$ 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 $m^{+}\not\equiv 0$
 and $\int_{\partial\Omega} m\,d\sigma<0$ 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.
  
Submitted August 10, 2004. Published August 14, 2005.
Math Subject Classifications: 35J70, 35P30.
Key Words: Nonlinear eigenvalue problem; Steklov problem; p-Laplacian;
           nonlinear boundary condition;  indefinite weight.