A positive bound state for an asymptotically linear or superlinear Schrödinger equation in exterior domains

Abstract

We establish the existence of a positive solution for semilinear elliptic equation in exterior domains \begin{document}$\begin{array}{lc}-Δ u + V(x) u = f(u), {\rm{in}} Ω \subseteq \mathbb{R}^N &&&(P_V)\end{array}$\end{document} where \begin{document}$N≥2$\end{document} , \begin{document}$Ω$\end{document} is an open subset of \begin{document}$\mathbb{R}^N$\end{document} and \begin{document}$ \mathbb{R}^N \setminus Ω $\end{document} is bounded and not empty but there is no restriction on its size, nor any symmetry assumption. The nonlinear term \begin{document}$f$\end{document} is a non homogeneous, asymptotically linear or superlinear function at infinity. Moreover, the potential Ⅴ is a positive function, not necessarily symmetric. The existence of a solution is established in situations where this problem does not have a ground state.