In this paper, we study the existence of solutions for the following superlinear elliptic equation with nonlinear boundary value condition $$ \left\{ \begin{ array}{ ll} -\Delta u+u=|u|^{ r-2}u &\text{ in} \; \Omega,\\ \\ \frac{ \partial u}{ \partial \nu}=|u|^{ q-2}u &\text{ on}\;\partial\Omega, \end{ array} \right. $$ where $\Omega\subset \mathbb R^N, N\geq 3$ is a bounded domain with smooth boundary. We will prove the existence results for the above equation under four different cases: (i) Both $q$ and $r$ are subcritical; (ii) $r$ is critical and $q$ is subcritical; (iii) $r$ is subcritical and $q$ is critical; (iv) Both $q$ and $r$ are critical.
Categories: math.AP