Ground states for the pseudo-relativistic Hartree equation with external potential

Abstract

We prove existence of positive ground state solutions to the pseudo-relativistic Schr\"{o}dinger equation \begin{equation*} \left\{ \begin{array}{l} \sqrt{-\Delta +m^2} u +Vu = \left( W * |u|^{\theta} \right)|u|^{\theta -2} u \quad\text{in $\mathbb{R}^N$}\\ u \in H^{1/2}(\mathbb{R}^N) \end{array} \right. \end{equation*} where $N \geq 3$, $m >0$, $V$ is a bounded external scalar potential and $W$ is a convolution potential, radially symmetric, satisfying suitable assumptions. We also furnish some asymptotic decay estimates of the found solutions.