### Neumann problems with double resonance

#### 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.

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.

#### Keywords

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

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