Ground state normalized solution to Schr\"odinger systems with general nonlinearities and potentials

In present paper, we prove the existence of solutions $(\lambda_1,\lambda_2, u_1,u_2)\in \R^2\times H^1(\R^N, \R^2)$ to systems of nonlinear Schrodinger equations with potentials $$\begin{cases} -\Delta u_1+V_1(x)u_1+\lambda_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{in}\;\R^N\\ -\Delta u_2+V_2(x)u_2+\lambda_2 u_2=\partial_2G(u_1,u_2)\;\quad&\hbox{in}\;\R^N\\ 00, ~ p_i, q_j>2, ~r_{1,k}, r_{2,k}>1. \end{cases} $$ Under some assumptions on $V_\iota$ and the parameters, we can prove the strict binding inequality for the mass sub-critical problem and obtain the existence of ground state normalized solutions for any given $a_1>0,a_2>0$.