Positive solutions for Choquard equation in exterior domains

Abstract

This work concerns with the following Choquard equation \begin{document}$ \begin{equation*} \begin{cases} -\Delta u+ u = (\int_{\Omega}\frac{u^2(y)}{|x-y|^{N-2}}dy)u &{\rm{in }}\; \Omega , u\in H_0^1(\Omega), \end{cases} \end{equation*} $\end{document} where \begin{document}$ \Omega\subseteq \mathbb{R}^{N} $\end{document} is an exterior domain with smooth boundary. We prove that the equation has at least one positive solution by variational and toplogical methods. Moreover, we establish a nonlocal version of global compactness result in unbounded domain.