STEKLOV PROBLEM WITH AN INDEFINITE WEIGHT FOR THE p-LAPLACIAN

Let Omega subset of R-N, with N >= 2, be a Lipschitz domain and let 1 < p < infinity. We consider the eigenvalue problem Delta(p)u = 0 in Omega and vertical bar del u vertical bar(p- 2) partial derivative u/partial derivative v = lambda m vertical bar u vertical bar p(-2)u on partial derivative Omega, where lambda is the eigenvalue and u is an element of W-1,W- 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 equivalent to 0 and integral partial derivative 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.