Multiple solutions for a class of nonlinear Neumann eigenvalue problems

Leszek Gasiński; Nikolaos S. Papageorgiou

doi:10.3934/cpaa.2014.13.1491

Article Contents

2014, Volume 13, Issue 4: 1491-1512. Doi: 10.3934/cpaa.2014.13.1491

This issue Previous Article Global strong solution to the two-dimensional density-dependent magnetohydrodynamic equations with vaccum Next Article Retraction

Multiple solutions for a class of nonlinear Neumann eigenvalue problems

Leszek Gasiński^1, and
Nikolaos S. Papageorgiou^2,

1.
Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków
2.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received: May 31, 2013

Revised: November 30, 2014

Published: June 2015

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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.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60, 35J92, 58E05.

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