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

Sergiu Aizicovici; Nikolaos S. Papageorgiou; V. Staicu

doi:10.3934/dcds.2009.25.431

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2009, Volume 25, Issue 2: 431-456. Doi: 10.3934/dcds.2009.25.431

This issue Previous Article Existence of five nonzero solutions with exact sign for a $p$-Laplacian equation Next Article Local rigidity of reducibility of analytic quasi-periodic cocycles on $U(n)$

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

Received: July 2008

Revised: December 2008

Published: June 2009

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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\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.

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Mathematics Subject Classification: Primary: 35J60, 35J65.

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