Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
157 tokens/sec
GPT-4o
8 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Limiting Behavior of Constraint Minimizers for Inhomogeneous Fractional Schrödinger Equations (2302.05834v3)

Published 12 Feb 2023 in math.AP

Abstract: This paper is devoted to the $L2$-constraint variational problem \begin{equation*} We study $L2$-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\beta2}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.

Summary

We haven't generated a summary for this paper yet.