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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

Sergiu Aizicovici 1, , Nikolaos S. Papageorgiou 2, and  V. Staicu 3,

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.
Keywords: index formula, homotopy invariant, crossing nonlinearity., $(S) _{+}-$operator, second eigenvalue.
Mathematics Subject Classification: Primary: 35J60, 35J6.
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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