In this paper, our aim is to prove the existence of normalized ground state for the following Schr\"odinger systems with potentials $$\begin{ cases} -\Delta u_1+V_1(x)u_1+\lambda_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{ in}\;\mathbb{ R}^N,\\ -\Delta u_2+V_2(x)u_2+\lambda_2 u_2=\partial_2G(u_1,u_2)\;\quad&\hbox{ in}\;\mathbb{ R}^N,\\ 0<u_1,u_2\in H^1(\mathbb{ R}^N), N\geq 1,\\ \int_{ \mathbb{ R}^N}u_1^2 \mathrm{ d} x=a_1, \int_{ \mathbb{ R}^N}u_2^2 \mathrm{ d} x=a_2. \end{ cases}$$ The potentials $V_1(x),V_2(x)$ are general such that $\inf \text{ ess}~\sigma(-\Delta+V_\iota)>-\infty$, which are allowed to be singular at some points. And the nonlinearities $G(u_1,u_2)$ are considered of the form $$ \begin{ cases} G(u_1, u_2):=\sum_{ i=1}^{ \ell}\frac{ \mu_i}{ p_i}|u_1|^{ p_i}+\sum_{ j=1}^{ m}\frac{ \nu_j}{ q_j}|u_2|^{ q_j}+\sum_{ k=1}^{ n}\beta_k |u_1|^{ r_{ 1,k}}|u_2|^{ r_{ 2,k}},~~\ell,m,n\in \mathbb{ N}^+_0, \mu_i, \nu_j,\beta_k>0, ~2<r_{ 1,k}+r_{ 2,k}, p_i, q_j<2+\frac{ 4}{ N}, ~r_{ 1,k}, r_{ 2,k}>1, i=1,2,\cdots, \ell; j=1,2,\cdots, m; k=1,2,\cdots, n. \end{ cases} $$ Under the mass sub-critical assumption, the normalized ground states are obtained as the minimum of the functional $J$ on the manifold $S_{ a_1,a_2}$. Since the functional is not weak lower semi-continuous, to prove the minimizing problem is achievable, the key step is establishing the strict sub-additive inequality. Among its main ingredients is the study of the sharp decay of the positive solutions and the interaction estimates.
Categories: math.AP