Resonance and the second BVP

Shapiro, Victor L.

doi:10.1090/S0002-9947-1991-0994172-7

Resonance and the second BVP
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by Victor L. Shapiro PDF

Trans. Amer. Math. Soc. 325 (1991), 363-387 Request permission

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: \[ Qu = \sum \limits _{1 \leq |\alpha | \leq m} {{{(- 1)}^{|\alpha |}}{D^\alpha }{A_\alpha }(x,{\xi _m}(u))}, \] where for $u \in {W^{m,p}}$, ${\xi _m}(u) = \{ {D^\alpha }u:|\alpha | \leq m\}$. (For $m = 1$, $Qu = - \sum \nolimits _{i = 1}^N {{A_i}(x,u,Du)}$.) Under four assumptions on ${A_\alpha }$—Carathéodory, growth, monotonicity for $|\alpha | = m$, 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 $\text {BVP}$.

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