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Transactions of the American Mathematical Society

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Resonance and the second BVP

Author: Victor L. Shapiro
Journal: Trans. Amer. Math. Soc. 325 (1991), 363-387
MSC: Primary 35J65
MathSciNet review: 994172
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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:

$\displaystyle Qu = \sum\limits_{1 \leq \vert\alpha \vert \leq m} {{{(- 1)}^{\vert\alpha \vert}}{D^\alpha }{A_\alpha }(x,{\xi _m}(u))}, $

where for $ u \in {W^{m,p}}$, $ {\xi _m}(u) = \{ {D^\alpha }u:\vert\alpha \vert \leq m\} $. (For $ m = 1$, $ Qu = - \sum\nolimits_{i = 1}^N {{A_i}(x,u,Du)}$.) Under four assumptions on $ {A_\alpha }$--Carathéodory, growth, monotonicity for $ \vert\alpha \vert = m$, and ellipticity--results at resonance are established for the equation $ Qu = G + f(x,u)$, where $ G \in {[{W^{m,p}}(\Omega)]^\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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Keywords: Quasilinear elliptic, resonance, generalized divergence form
Article copyright: © Copyright 1991 American Mathematical Society