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Journal Article

# Resonance and the Second BVP

Victor L. Shapiro
Transactions of the American Mathematical Society
Vol. 325, No. 1 (May, 1991), pp. 363-387
DOI: 10.2307/2001675
Stable URL: http://www.jstor.org/stable/2001675
Page Count: 25

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## Abstract

Let $\Omega \subset \mathbb{R}^N$ be a bounded open connected set with the cone property, and let $1 < p < \infty$. Also, let $Qu$ be the $2m$th order quasilinear differential operator in generalized divergence form: $$Qu = \sum_{1\leq|\alpha|\leq m} (-1)^{|\alpha|} D^\alpha A_\alpha(x, \xi_m(u)),$$ where for $u \in W^{m,p}, \xi_m(u) = \{D^\alpha u:|\alpha| \leq m\}$. (For $m = 1, Qu = - \sum_{i=1}^N A_i(x, u, Du)$.) Under four assumptions on $A_\alpha$--Caratheodory, growth, monotonicity for $|\alpha| = m$, and ellipticity--results at resonance are established for the equation $Qu = G + f(x, u)$, where $G \in \lbrack W^{m,p}(\Omega) \rbrack^\ast$ and $f(x, u)$ satisfies a one-sided condition (plus others). For the case $m = 1$, these results are tantamount to generalized solutions of the second BVP.

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