Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schr\"{o}dinger Equations

Abstract

This paper is devoted to the $L^2$-constraint variational problem \begin{equation*} We study $L^2$-normalized solutions of the following inhomogeneous fractional Schr\"{o}dinger equation \begin{equation*} (-\Delta)^{s} u(x)+V(x)u(x)-a|x|^{-b}|u|^{2\beta^2}u(x)=\mu u(x)\ \ \mbox{in}\ \ \R^{N}. \end{equation*} Here $s\in(\frac{1}{2},1)$, $N>2s$, $a>0$, $0 0$ such that minimizers exist for $0 a^*$. In the case of $a=a^*$, one can obtain the classification results of the existence and non-existence for constraint minimizers, which are depended strongly on the value of $V(0)$. For $V(0)=0$, the limiting behavior of nonnegative minimizers is also analyzed when $a$ tend to $a^*$ from below.