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

Given a connected open set 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 we prove that the positive principal eigenvalue of the problem 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. Received: 18 March 2005; revised: 7 April 2005