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

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 \(w^{+} \not\equiv 0,\) 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.