Existence of solution for Schr\"odinger-Newton system with vanishing magnetic potential

Abstract

In this paper, we study the following Schr\"odinger-Newton system with a vanishing magnetic potential and an electric potential \begin{equation}\label{0.1} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} (-i\nabla +A(x))^2u+V(x)u=\Phi u, \&x\in\mathbb{R}^3,\\ -\Delta \Phi =|u|^2, \&x\in\mathbb{R}^3, \end{array} \right. \end{equation} where $i$ is the imaginary unit, $A:\mathbb{R}^3\to \mathbb{R}^3$ is a continuous magnetic potential, and $V:\mathbb{R}^3\to \mathbb{R}$ is a continuous electric potential. First, we show that the ground state energy of system with $A(x)\equiv0$ and $V(x)\equiv V_{\infty}>0$ is isolated. Using this fact together with a global compactness lemma, a barycenter function, and Brouwer degree theory, we prove that under suitable conditions on $A$ and $V$, system admits either a ground state solution or a high energy solution. Interestingly, whether a ground state solution exists is significantly affected by the magnetic potential $A$ and the electric potential $V$. Furthermore, we provide sufficient conditions that guarantee the existence of a ground state solution, as well as conditions under which no ground state solution exists and a high energy solution appears instead.