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

TL;DR

This paper investigates the existence and behavior of solutions to an inhomogeneous fractional Schrödinger equation under $L^2$ constraints, identifying a critical parameter value that determines the existence of minimizers and analyzing their limiting behavior.

Contribution

It establishes the existence of a critical parameter for minimizers of the constrained problem and classifies solutions based on the potential's value at zero, including their limiting behavior.

Findings

Existence of minimizers for $0<a<a^*$

Non-existence for $a>a^*$

Limiting behavior of minimizers as $a$ approaches $a^*$ from below

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 $s\in(\frac{1}{2},1)$, $N>2s$, $a>0$, $0<b<\min\{\frac{N}{2},1\}$, $\beta=\sqrt{\frac{2s-b}{N}}$ and $V(x)\geq 0$ 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 $a^*>0$ such that minimizers exist for $0<a<a^*$, and minimizers do not exist for any $a>a^*$. 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.