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

Abstract:

We consider the Neumann problem \[ \left \{ {\begin {array}{*{20}{c}} { - \Delta u = g(u) + h(x){\text { in }}\Omega ,} \\ {\partial u/\partial \nu = 0\quad {\text {on }}\operatorname {bdry} \Omega .} \\ \end {array} } \right .\] 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 $\mathbb {R}$.