Positive Solutions in Exterior Domains of Nonlinear Field Equations

We establish the existence of a positive solution to the problem
\[
-\Delta u+V(x)u=f(u),\qquad x \in \Omega \subset \mathbb{R}^{N},
\qquad u(x) \to 0\;\;\text{as}\;\; |x| \to \infty,
\]
for \(N\geq3\), \(\Omega\) a regular unbounded exterior domain, when either the nonlinearity \(f\) is subcritical and superquadratic or asymptotically linear at infinity, in case \(V\) approaches a positive constant limit at infinity, or \(f\) is subcritical at infinity and
supercritical near the origin if the potential \(V\) vanishes at infinity.
Under a suitable decay assumption on the potential \(V\), we show that the
problem has a positive bound state, including situations in which the problem does not have a ground state.
This is work in collaboration with A. Khatib (UnB, Brazil).