Sharp interaction estimates and their application: existence of normalized ground states to coupled Schr\"odinger systems with potentials

Abstract

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.