c©2006, Sociedade Brasileira de Matemática ON THE EXISTENCE OF SIGNED SOLUTIONS FOR A QUASILINEAR ELLIPTIC PROBLEM IN RN

We study the existence of positive and negative weak solutions for the equation −∆pu+ V (x)|u| p−2u = λf(u) in RN, where −∆pu = div(|∇u| p−2∇u) is the p-Laplacian operator, 1 < p < N, λ is a positive real parameter and the potential V: RN → R is bounded from below for a positive constant and ”large ” at infinity. It is assumed that the nonlinearity f: R → R is continuous and just superlinear in a neighborhood of the origin.