Topological Methods in Nonlinear Analysis

Neumann problems with double resonance

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

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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.

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

First available in Project Euclid: 20 April 2016

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O'Regan, Donal; Papageorgiou, Nikolaos S.; Smyrlis, George. Neumann problems with double resonance. Topol. Methods Nonlinear Anal. 39 (2012), no. 1, 151--173.

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