The spectrum and an index formula for the Neumann p-Laplacian and multiple solutions for problems with a crossing nonlinearity

Abstract

In this paper we first conduct a study of the spectrum of the negative p-Laplacian with Neumann boundary conditions. More precisely we investigate the first nonzero eigenvalue. We produce alternative variational characterizations, we examine its dependence on p 6 (l,∞) and on the weight function m ∈ L∞ (Z)+ and we prove that the isolation of the principal eigenvalue λo = 0, is uniform for all p in a bounded closed interval. All these results are then used to prove an index formula (jumping theorem) for the d(s)+-degree map at the first nonzero eigenvalue. Finally the index formula is used to prove a multiplicity result for problems with a multivalued crossing nonlinearity.