Normalized solutions for pseudo-relativistic Schrödinger equations

Xueqi Sun; Yongqiang Fu; Sihua Liang; Xueqi Sun; Yongqiang Fu; Sihua Liang

doi:10.3934/cam.2024010

Communications in Analysis and Mechanics

2024, Volume 16, Issue 1: 217-236. doi: 10.3934/cam.2024010

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Normalized solutions for pseudo-relativistic Schrödinger equations

1.
College of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China
2.
College of Mathematics, Changchun Normal University, Changchun, 130032, P.R. China

Received: 08 September 2023 Revised: 28 January 2024 Accepted: 01 February 2024 Published: 19 February 2024
35J20, 35R03, 58E05

In this paper, we consider the existence and multiplicity of normalized solutions to the following pseudo-relativistic Schrödinger equations

$\begin{equation*} \left\{ \begin{array}{lll} \sqrt{-\Delta+m^2}u +\lambda u = \vartheta |u|^{p-2}v +|u|^{2^\sharp-2}v, & x\in \mathbb{R}^N, \ u>0, \\ \ \int_{{\mathbb{R}^N}}|u|^2dx = a^2, \end{array} \right. \end{equation*}$

where $N\geq2,$ $a, \vartheta, m > 0,$ $\lambda$ is a real Lagrange parameter, $2 < p < 2^\sharp = \frac{2N}{N-1}$ and $2^\sharp$ is the critical Sobolev exponent. The operator $\sqrt{-\Delta+m^2}$ is the fractional relativistic Schrödinger operator. Under appropriate assumptions, with the aid of truncation technique, concentration-compactness principle and genus theory, we show the existence and the multiplicity of normalized solutions for the above problem.

Keywords:

pseudo-relativistic Schrödinger operator,
normalized solutions,
Sobolev critical exponent

Citation: Xueqi Sun, Yongqiang Fu, Sihua Liang. Normalized solutions for pseudo-relativistic Schrödinger equations[J]. Communications in Analysis and Mechanics, 2024, 16(1): 217-236. doi: 10.3934/cam.2024010

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Abstract

In this paper, we consider the existence and multiplicity of normalized solutions to the following pseudo-relativistic Schrödinger equations

A correction on

Monotonicity and inequalities related to complete elliptic integrals of the second kind

by Fei Wang, Bai-Ni Guo and Feng Qi. AIMS Mathematics, 2020, 5(3): 2732–2742.

DOI: 10.3934/math.2020176

In Acknowledgments section, the Grant number of "Project for Combination of Education and Research Training at Zhejiang Institute of Mechanical and Electrical Engineering" is missing. Here we give the complete information of this fund.

The changes have no material impact on the conclusion of this article. The original manuscript will be updated ^[1]. We apologize for any inconvenience caused to our readers by this change.

Acknowledgments

This work was partially supported by the Foundation of the Department of Education of Zhejiang Province (Grant No. Y201635387), the National Natural Science Foundation of China (Grant No. 11171307), the Visiting Scholar Foundation of Zhejiang Higher Education (Grant No. FX2018093), and the Project for Combination of Education and Research Training at Zhejiang Institute of Mechanical and Electrical Engineering (Grant No. A027120206).

The authors thank anonymous referees for their careful corrections to, helpful suggestions to, and valuable comments on the original version of this manuscript.

Conflict of interest

The authors declare that they have no conflict of interest.

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Communications in Analysis and Mechanics

Normalized solutions for pseudo-relativistic Schrödinger equations

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Normalized solutions for pseudo-relativistic Schrödinger equations

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