Normalized solutions to fractional Schrödinger equation with potentials

Mei-Qi Liu; Wenming Zou

doi:10.3934/dcdss.2023188

Article Contents

2023, Volume 16, Issue 11: 3194-3211. Doi: 10.3934/dcdss.2023188

This issue Previous Article On multi-bump solutions for the Choquard-Kirchhoff equations in $ \mathbb{R}^{N} $ Next Article On the higher order mean curvatures of spacelike hypersurfaces in pp-wave spacetimes

Normalized solutions to fractional Schrödinger equation with potentials

Mei-Qi Liu and
Wenming Zou^,

Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China

^*Corresponding author: Wenming Zou
^*Corresponding author: Wenming Zou

Received: May 2023

Revised: September 2023

Early access: October 10, 2023

Published online: October 10, 2023

This work is supported by NSFC-12171265.

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Abstract

In this paper, we utilize variational method to study the following fractional Schrödinger equation with a prescribed $ L^2 $-mass:

$ \begin{equation*} \begin{cases} (-\Delta)^s u+(V(x)+\lambda)u = g(u)\;\; \hbox{in}\;\mathbb{R}^N,\\ \int_{\mathbb R^N}|u|^2 \mathrm{d} x = a. \end{cases} \end{equation*} $

Here $ (-\Delta)^s $ is the fractional Laplacian operator, $ s\in(0,1) $, $ N\geq2 $, $ a>0 $, and $ \lambda\in\mathbb R $ will arise as a Lagrange multiplier. Under some suitable assumptions on $ V $ and $ g $, we develop an interesting method based on iterative techniques to deduce the sub-additive inequality about the $ L^2 $-constraint minimization problem. Furthermore, we obtain the existence of the normalized solution, which extends the results of [Shibata, in Manuscripta Math 143: 221-237, 2014] and [Cingolani et al. in Nonlinearity 34: 4017-4056, 2021].

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Mathematics Subject Classification: Primary: 26A33, 35J20; Secondary: 35Q55.

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