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Proceedings of the American Mathematical Society

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A necessary and sufficient condition of nonresonance for a semilinear Neumann problem


Authors: J.-P. Gossez and P. Omari
Journal: Proc. Amer. Math. Soc. 114 (1992), 433-442
MSC: Primary 35J65; Secondary 47H15
DOI: https://doi.org/10.1090/S0002-9939-1992-1091181-3
MathSciNet review: 1091181
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Abstract: We consider the Neumann problem

$\displaystyle \left\{ {\begin{array}{*{20}{c}} { - \Delta u = g(u) + h(x){\text... ...\nu = 0\quad {\text{on }}\operatorname{bdry} \Omega .} \\ \end{array} } \right.$

Assuming some growth restriction on the nonlinearity $ g$, we prove that a necessary and sufficient condition for the existence of a solution for every given $ h \in {L^\infty }(\Omega )$ is that $ g$ be unbounded from above and from below on $ \mathbb{R}$.

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DOI: https://doi.org/10.1090/S0002-9939-1992-1091181-3
Article copyright: © Copyright 1992 American Mathematical Society