Communications on Pure and Applied Analysis (CPAA)

Multiple solutions for a class of nonlinear Neumann eigenvalue problems

Pages: 1491 - 1512, Volume 13, Issue 4, July 2014      doi:10.3934/cpaa.2014.13.1491

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Leszek Gasiński - Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków, Poland (email)
Nikolaos S. Papageorgiou - Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece (email)

Abstract: We consider a parametric nonlinear equation driven by the Neumann $p$-Laplacian. Using variational methods we show that when the parameter $\lambda > \widehat{\lambda}_1$ (where $\widehat{\lambda}_1$ is the first nonzero eigenvalue of the negative Neumann $p$-Laplacian), then the problem has at least three nontrivial smooth solutions, two of constant sign (one positive and one negative) and the third nodal. In the semilinear case (i.e., $p=2$), using Morse theory and flow invariance argument, we show that the problem has three nodal solutions.

Keywords:  Smooth solutions of constant sign, nodal solutions, first nonzero eigenvalue, mountain pass theorem, critical groups, negative gradient flow, flow invariance.
Mathematics Subject Classification:  Primary: 35J20; Secondary: 35J60, 35J92, 58E05.

Received: June 2013;      Revised: December 2014;      Available Online: February 2014.