# American Institute of Mathematical Sciences

July  2009, 25(2): 431-456. doi: 10.3934/dcds.2009.25.431

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

 1 Department of Mathematics, Ohio University, Athens, OH 45701 2 Department of Mathematics, National Technical University, Zografou Campus, Athens 15780 3 Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

Received  July 2008 Revised  December 2008 Published  June 2009

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 $p\in( 1,\infty)$ and on the weight function $m\in L^{\infty}(Z) _{+}$ and we prove that the isolation of the principal eigenvalue $\lambda_{0}=0,$ 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.
Citation: Sergiu Aizicovici, Nikolaos S. Papageorgiou, V. Staicu. The spectrum and an index formula for the Neumann $p-$Laplacian and multiple solutions for problems with a crossing nonlinearity. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 431-456. doi: 10.3934/dcds.2009.25.431
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