Neumann problems with double resonance

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


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.


Double resonance; C-condition; unique continuation property; critical groups; Morse theory; homotopy invariance

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