Multiple solutions for a class of nonlinear Neumann eigenvalue problems

Abstract

We consider a parametric nonlinear equation driven by the Neumann p-Laplacian. Using variational methods we show that when the parameter λ > λ̂1 (where λ̂1 is the first nonzero eigenvalue of the negative Neumann p-Laplacian), then the problem has at least three nontrivial smooth solutions, two of constant sign (one positive and one negative) and the third nodal. In the semilinear case (i.e., p = 2), using Morse theory and flow invariance argument, we show that the problem has three nodal solutions.