Exterior nonlinear Neumann problem (Q2644281)

Let \(\Omega\subset\mathbb R^N\), \(N\geq 3\) be a bounded domain with smooth boundary and it is assumed that \(\Omega^c= \mathbb R^N\setminus\Omega\) does not have bounded components. The authors consider on \(\Omega^c\) the exterior Neumann problem \[ \begin{gathered} -\Delta u+ b(x)u= Q(x)|u|^{p-2} u\quad\text{in }\Omega^c,\;u> 0\quad\text{on }\Omega^c,\\ {\partial u\over\partial\nu}= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(2< p\leq 2^*\), \(2^*:= {2N\over N-2}\) and \(\nu\) is the outer normal to \(\partial\Omega\). They deal with both subcritical and critical cases. They establish a criterion for the existence of least energy solutions of (1). The authors show that least energy solutions exhibit a tendency to concentrate to a solution of a nonlinear problem with mixed boundary value conditions.