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} &-Δu+λ_1 u+V_1(x)u=μ_1(I_α\star|u|^p)|u|^{p-2}u+β(x) v,\\ &-Δv+λ_2 v+V_2(x)v=μ_2(I_α\star|v|^q)|v|^{q-2}u+β(x) u, \end{aligned} \right.\quad x\in \mathbb{R}^N, \end{align*} under the constraint \begin{align*} \int_{\mathbb{R}^N}u^2dx=ξ^2,~ \int_{\mathbb{R}^N}v^2dx=η^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.