{'fraction_non_alphanumeric': 0.057660626029654036, 'fraction_numerical': 0.006589785831960461, 'mean_word_length': 5.204081632653061, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Variational Eigenvalues of Degenerate Eigenvalue Problems for the Weighted p-Laplacian

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 L1loc(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.