Variational Eigenvalues of Degenerate Eigenvalue Problems for the Weighted p-Laplacian
An Lê, Klaus Schmitt · Advanced Nonlinear Studies · 2005
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-2 u, λ > 0 subject to Dirichlet boundary conditions on a bounded domain . The diffusion coefficient a(x) is a function in L 1 loc (Ω) and b(x) is a nontrivial function in L r (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.