  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.
