On exterior Neumann problems with an asymptotically linear nonlinearity (Q996237)

The author studies the following semilinear elliptic problem in an exterior domain with the Neumann boundary condition \[ \begin{gathered} -\Delta u+ u+ f(u)\quad\text{in }\mathbb R^N\setminus \overline\Omega,\\ {\partial u\over\partial\nu}= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(\Omega\subset\mathbb R^N\) is an open bounded domain with \(\partial\Omega\in C^1\), \(N\geq 3\), and \(\nu\) is the interior unit normal vector on \(\partial\Omega\). The author is interested in the existence of a ground state solution of (1) and under the natural assumptions on \(f\) proves its existence. Moreover, the author shows that, when \(\Omega\) is a ball, a ground state solution not necessarily radially symmetric and gives also asymptotic profile of ground state solutions.