Concentration Behavior of Ground States for $L^2$-Critical Schr\"{o}dinger Equation with a Spatially Decaying Nonlinearity

TL;DR

This paper studies the existence and concentration behavior of ground states for an $L^2$-critical Schr"odinger equation with spatially decaying nonlinearity, revealing a threshold phenomenon and detailed asymptotic analysis near criticality.

Contribution

It establishes a threshold for the existence of minimizers and analyzes their concentration behavior as the parameter approaches this threshold.

Findings

Existence of a critical threshold $a^*$ for minimizers.

Precise concentration behavior of minimizers as $a$ approaches $a^*$.

Uniqueness of minimizers at the threshold under certain conditions.

Abstract

We consider the following time-independent nonlinear $L^2$-critical Schr\"{o}dinger equation \[ -\Delta u(x)+V(x)u(x)-a|x|^{-b}|u|^{1+\frac{4-2b}{N}}=\mu u(x)\,\ \hbox{in}\,\ \mathbb{R}^N, \] where $\mu\in\mathbb{R}$, $a>0$, $N\geq 1$, $0<b<\min\{2,N\}$, and $V(x)$ is an external potential. It is shown that ground states of the above equation can be equivalently described by minimizers of the corresponding minimization problem. In this paper, we prove that there is a threshold $a^*>0$ such that minimizer exists for $0<a<a^*$ and minimizer does not exist for any $a>a^*$. However if $a=a^*$, it is proved that whether minimizer exists depends sensitively on the value of $V(0)$. Moreover, when there is no minimizer at threshold $a^*$, we give a detailed concentration behavior of minimizers as $a\nearrow a^*$, based on which we finally prove that there is a unique minimizer as $a\nearrow a^*$.