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

arXiv2302.05834MaRDI QIDQ6508791

Hongfei Zhang, Shu Zhang

Abstract: This paper is devoted to studying the following fractional $L^2$-critical nonlinear Schr"odinger equation ${\displaystyle }$(-Delta)^{s} u(x)+V(x)u(x)-a|x|^{-b}|u|^{frac{4s-2b}{N}}u(x) = mu u(x), hbox{in}, mathbb{R}^N,${\displaystyle }$ where $muinmathbbR$, $a>0$, $sin(frac12,1)$, $N>2s$, $0<b<min2s,fracN2$ and $V(x)geq0$ is an external potential. We obtain normalized $L^2$-norm solutions of the above equation by solving the associated constraint minimization problem (1.4). It shows that there is a threshold $a^{*}>0$ such that (1.4) has minimizers for $0<a<a^{*}$, and minimizers do not exist for any $a>a^{*}$. For the case of $a=a^{*}$, it gives a fact that the existence and non-existence of minimizers depend strongly on the value of $V(0)$. Especially for $V(0)=0$, we prove that minimizers occur blow-up behavior and the mass of minimizers concentrates at the origin as $aearrow{a}^{*}$. Applying implicit function theorem, the uniqueness of minimizers is also proved for $a>0$ small enough.

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