Variational Eigenvalues of Degenerate Eigenvalue Problems for the Weighted p-Laplacian

Abstract

Abstract We prove the existence of nondecreasing sequences of positive eigenvalues of the homogeneous degenerate quasilinear eigenvalue problem − div(a(x)|▽u|p-2▽u) = λb(x)|u|p-2u, λ > 0 subject to Dirichlet boundary conditions on a bounded domain . The diffusion coefficient a(x) is a function in L1 loc(Ω) and b(x) is a nontrivial function in Lr(Ω) (r depending on a, p and N) and may change sign. We use Ljusternik-Schnirelman theory, minimax theory and the theory of weighted Sobolev spaces to establish our results.