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.
- 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

$ \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.

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