On some general semilinear elliptic problems with nonlinear boundary conditions (Q1575804)

The authors consider the following boundary value problem: \(\Delta u=p(u)\) in \(\Omega\), \(u=0\) on \(\Gamma_0\), \(\partial u/\partial n =q(u)\) on \(\Gamma_1\), where \(\Omega\) is a smooth bounded domain in \({\mathbb R}^n\) whose boundary is split in two parts \(\Gamma_0\) and \(\Gamma_1\). In the case \(n=1\), it is found a necessary and sufficient condition for the existence of a solution. Moreover, a multiplicity result is obtained. In the case \(n>1\), the authors prove some existence results for a wide class of nonlinearities.