Existence results for superlinear elliptic equations with nonlinear boundary value conditions

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.