We consider the problem $-\text{div}(p(x)\nabla u)=\lambda{u}+\alpha|u|^{r-1}u$ in $\varOmega$, $\partial u/\partial\nu=Q(x)|u|^{q-2}u$ on $\partial\varOmega$, where $\varOmega$ is a bounded smooth domain in $\mathbb{R}^{N}$, $N\geq3$, $q=2(N-1)/(N-2)$ and $2<r<q$. Under some conditions on $\partial\varOmega$, $p$, $Q$, $\lambda$, $\alpha$ and the mean curvature at some point $x_0$, we prove the existence of solutions of the above problem. We use variational arguments, namely the concentration–compactness principle, min–max principle and the mountain-pass theorem.