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

Abstract

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