A Landesman-Lazer type condition for a nonlinear Stekloff problem (Q2474197)

The paper deals with the nonlinear Stekloff problem \[ \Delta u=f\;\text{ in } \Omega,\qquad u=0\;\text{ on }\gamma,\qquad \frac{\partial u}{\partial\nu}= \mu u+\beta(u)+g\;\text{ on }\Gamma. \] Here \(\Omega\subset\mathbb R^n\) is a bounded domain with smooth boundary \(\partial\Omega=\Gamma\cup\gamma,\) \(\Gamma\cap\gamma=\emptyset,\) \(\gamma\neq\emptyset.\) Existence of a unique solution is proved when \(\mu\) is not an eigenvalue of the corresponding linear problem. Necessary and sufficient conditions (similar to the classical Landesman-Lazer ones) for the existence of solutions are obtained in the resonant case \(\mu=\mu_k\) by virtue of Lyapunov-Schmidt procedure.