We study the bifurcation problem

$$ \begin{cases} -\Div(a(x)|Du|^{p-2}Du)+h(x)u^{r-1}= f(\lambda,x,u) & \text{in } \sOm\subset\RR^{N}, \\ a(x)|Du|^{p-2}Du\cdot n+b(x)u^{p-1}=\theta g(x,u) & \text{on }\sGa, \\ u\geq0,\quad u\not\equiv0 & \text{in }\sOm, \end{cases} $$

where $\sOm$ is an unbounded domain with smooth non-compact boundary $\sGa$, $n$ denotes the unit outward normal vector on $\sGa$, and $\lambda>0$, $\theta$ are real parameters. We assume throughout that $p lt r lt p^{*}=pN/ (N-p)$, $1 lt p lt N$, the functions $a$, $b$ and $h$ are positive while $f$, $g$ are subcritical nonlinearities. We show that there exist an open interval $I$ and $\lambda^\star gt 0$ such that the problem has no solution if $\theta\in I$ and $\lambda\in (0,\lambda^\star)$. Furthermore, there exist an open interval $J\subset I$ and $\lambda_0 gt 0$ such that, for any $\theta\in J$, the above problem has at least a solution if $\lambda\geq \lambda_0$, but it has no solution provided that $\lambda\in (0,\lambda_0)$.

AMS 2000 Mathematics subject classification: Primary 35J60; 35P30; 58E05; 58G28