Mass Concentration and Local Uniqueness of Ground States for $L^2$-subcritical Nonlinear Schr��dinger Equations

We consider ground states of $L^2$-subcritical nonlinear Schr��dinger equation (1.1), which can be described equivalently by minimizers of the following constraint minimization problem $$ e(��):=\inf\{E_��(u):u\in \mathcal{H}(\mathbb{R}^d),\|u\|_2^2=1\}.$$ The energy functional $E_��(u)$ is defined by $$ E_��(u):=\frac{1}{2}\int_{\mathbb{R}^d}|\nabla u|^2dx +\frac{1}{2}\int_{\mathbb{R}^d}V(x)|u|^2dx-\frac{��^{p-1}}{p+1}\int_{\mathbb{R}^d}|u|^{p+1}dx,$$ where $d\geq1$, $��>0$, $p\in\big(1, 1+\frac{4}{d}\big)$ and $0\leq V(x)\to\infty$ as $|x| \to\infty$. We present a detailed analysis on the concentration behavior of ground states as $��\to\infty$, which extends the concentration results shown in [22]. Moreover, the uniqueness of nonnegative ground states is also proved when $��$ is large enough.