Normalized solutions of linear and nonlinear coupled Choquard systems with potentials

In this paper, we study Choquard systems with linear and nonlinear couplings with different potentials under the \(L^2\) $L^2$ -constraint. We use Ekland variational principle to prove this system has a normalized radially symmetric solution for \(L^2\) $L^2$ -subcritical case when the dimension is greater than or equal to 2 without potentials. In addition, a positive solution with prescribed \(L^2\) $L^2$ -constraint under some appropriate assumptions with the potentials was obtained. The proof is based on the refined energy estimates.