Topological Methods in Nonlinear Analysis

Neumann problems with double resonance

Donal O'Regan, Nikolaos S. Papageorgiou, and George Smyrlis

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Abstract

We study elliptic Neumann problems in which the reaction term at infinity is resonant with respect to any pair $\{ \widehat{\lambda}_m, \widehat{\lambda}_{m+1}\}$ of distinct consecutive eigenvalues. Using variational methods combined with Morse theoretic techniques, we show that when the double resonance occurs in a ``nonprincipal'' spectral interval $[\widehat{\lambda}_m, \widehat{\lambda}_{m+1}]$, $ m\geq 1$, we have at least three nontrivial smooth solutions, two of which have constant sign. If the double resonance occurs in the ``principal'' spectral $[\widehat{\lambda}_0=0,\widehat{\lambda}_1]$, then we show that the problem has at least one nontrivial smooth solution.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 39, Number 1 (2012), 151-173.

Dates
First available in Project Euclid: 20 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461184859

Mathematical Reviews number (MathSciNet)
MR2952308

Zentralblatt MATH identifier
1273.35145

Citation

O'Regan, Donal; Papageorgiou, Nikolaos S.; Smyrlis, George. Neumann problems with double resonance. Topol. Methods Nonlinear Anal. 39 (2012), no. 1, 151--173. https://projecteuclid.org/euclid.tmna/1461184859


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