Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schrödinger 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 Schrödinger equation \begin{equation*} (-Δ)^{s} u(x)+V(x)u(x)-a|x|^{-b}|u|^{2β^2}u(x)=μu(x)\ \ \mbox{in}\ \ \R^{N}. \end{equation*} Here $s\in (\frac{1}{2},1)$, $N>2s$, $a>0$, $0<b<\min \{\frac{N}{2},1\}$, ${\hat{I}}^2=\sqrt{\frac{2s-b}{N}}$ and $V(x)\ge 0$ is an external potential. We get $L^2$-normalized solutions of the above equation by solving the associated constrained minimization problem. We prove that there exists a critical value $a^{\ast }>0$ such that minimizers exist for $0<a<a^{\ast }$, and minimizers do not exist for any $a>a^{\ast }$. In the case of $a=a^{\ast }$, 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 $a^{\ast }$ from below.