Normalized solutions of linearly coupled Choquard system with potentials

Abstract

In this paper, we consider the existence of solutions for the linearly coupled Choquard system with potentials \begin{align*} \left\{\begin{aligned} &-\Delta u+\lambda_1 u+V_1(x)u=\mu_1(I_{\alpha}\star|u|^p)|u|^{p-2}u+\beta(x) v,\\ &-\Delta v+\lambda_2 v+V_2(x)v=\mu_2(I_{\alpha}\star|v|^q)|v|^{q-2}u+\beta(x) u, \end{aligned} \right.\quad x\in \mathbb{R}^N, \end{align*} under the constraint \begin{align*} \int_{\mathbb{R}^N}u^2dx=\xi^2,~ \int_{\mathbb{R}^N}v^2dx=\eta^2, \end{align*} where $I_{\alpha}=\frac{1}{|x|^{N-\alpha}},~\alpha\in(0,N),~1+\frac{\alpha}{N}<p,~q<\frac{N+\alpha}{N-2},~\mu_1>0,~\mu_2>0$ and $\beta(x)$ is a fixed function.