Sharp interaction estimates and their application: existence of normalized ground states to coupled Schrödinger systems with potentials
Yinbin Deng, Qihan He, Xuexiu Zhong
Abstract
In this paper, our aim is to prove the existence of normalized ground state for the following Schrödinger systems with potentials $$\begin{cases} -Δu_1+V_1(x)u_1+λ_1 u_1=\partial_1 G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\ -Δu_2+V_2(x)u_2+λ_2 u_2=\partial_2G(u_1,u_2)\;\quad&\hbox{in}\;\mathbb{R}^N,\\ 0<u_1,u_2\in H^1(\mathbb{R}^N), N\geq 1,\\ \int_{\mathbb{R}^N}u_1^2 \mathrm{d} x=a_1, \int_{\mathbb{R}^N}u_2^2 \mathrm{d} x=a_2. \end{cases}$$ The potentials $V_1(x),V_2(x)$ are general such that $\inf \text{ess}~σ(-Δ+V_ι)>-\infty$, which are allowed to be singular at some points. And the nonlinearities $G(u_1,u_2)$ are considered of the form $$ \begin{cases} G(u_1, u_2):=\sum_{i=1}^{\ell}\frac{μ_i}{p_i}|u_1|^{p_i}+\sum_{j=1}^{m}\frac{ν_j}{q_j}|u_2|^{q_j}+\sum_{k=1}^{n}β_k |u_1|^{r_{1,k}}|u_2|^{r_{2,k}},~~\ell,m,n\in \mathbb{N}^+_0, μ_i, ν_j,β_k>0, ~2<r_{1,k}+r_{2,k}, p_i, q_j<2+\frac{4}{N}, ~r_{1,k}, r_{2,k}>1, i=1,2,\cdots, \ell; j=1,2,\cdots, m; k=1,2,\cdots, n. \end{cases} $$ 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.