Nodal domains for the p-Laplacian

In this paper we consider the eigenvalue problem −Δ_pu = λ(m)|u|^p−2u, u ∈ W₀^1,p (Ω) where p > 1, Δ_p is the p-Laplacian operator, λ > 0, Ω is a bounded domain in ℝ^N (N ≥ 1) and m is a given positive function in L^r (Ω) (r depending on p and N). We prove that the second positive eigenvalue admits exactly two nodal domains.