On a nonlinear elliptic boundary value problem

Abstract:

Consider a bounded domain $G \subset {R^N}(N \geq 1)$ with smooth boundary $\Gamma$. Let $L$ be a uniformly elliptic linear differential operator. Let $\gamma$ and $\beta$ be two maximal monotone mappings in $R$. We prove that, when $\gamma$ satisfies a certain growth condition, given $f \in {L^2}(G)$ there is $u \in {H^2}(G)$ such that \[ Lu + \gamma (u) \backepsilon f\quad {\text {a.}}{\text {e.}}{\text { on }}G,\quad {\text {and}}\quad - \partial u/\partial v \in \beta ({u_{|\Gamma }})\quad {\text {a.}}{\text {e.}}{\text { on }}\Gamma ,\] where $\partial u/\partial v$ is the conormal derivative associated with $L$.