Simplicity of principal eigenvalue for p-Laplace operator with singular indefinite weight

摘要

Given a connected open set $Omega subset mathbb{R}^{N} $ and a function w ∈L N/p (Ω) if 1 < p < N and w ∈L r (Ω) for some r ∈(1, ∞) if p ≧ N, with $w^{+} notequiv 0,$ we prove that the positive principal eigenvalue of the problem $$ - hbox{div}(|nabla _{u} |^{{p - 2}} nabla u) = lambda w(x)|u|^{{p - 2}} u,quad u in mathcal{D}^{{1,p}}_{0} (Omega ), $$ is unique and simple. This improves previous works all of which assumed w in a smaller space than L N/p (Ω) to ensure that Harnack’s inequality holds. Our proof does not rely on Harnack’s inequality, which may fail in our case.