Existence and multiplicity of solutions for Neumann problems

Abstract

In this paper we examine semilinear and nonlinear Neumann problems with a nonsmooth locally Lipschitz potential function. Using variational methods based on the nonsmooth critical point theory, for the semilinear problem we prove a multiplicity result under conditions of double resonance at higher eigenvalues. Our proof involves a nonsmooth extension of the reduction method due to Castro–Lazer–Thews. The nonlinear problem is driven by the p-Laplacian. So first we make some observations about the beginning of the spectrum of ( − Δ p , W 1 , p ( Z ) ) . Then we prove an existence and multiplicity result. The existence result permits complete double resonance. The multiplicity result specialized in the semilinear case (i.e. p = 2 ) corresponds to the super-sub quadratic situation.