Open Access
June, 2024 Multiplicity of Normalized Solutions for Schrödinger Equation with Mixed Nonlinearity
Lin Xu, Changxiu Song, Qilin Xie
Author Affiliations +
Taiwanese J. Math. 28(3): 589-609 (June, 2024). DOI: 10.11650/tjm/240202

Abstract

In this paper, we explore the multiplicity of normalized solutions for Schrödinger equation with mixed nonlinearities \[ \begin{cases} -\Delta u + V(\epsilon x)u = \lambda u + \mu |u|^{q-2} u + |u|^{p-2} u &\textrm{in $\mathbb{R}^{N}$}, \\ \int_{\mathbb{R}^{N}} |u|^{2} \, dx = c, \end{cases} \] where $\mu \gt 0$, $c \gt 0$, $2 \lt q \lt 2+4/N \lt p \lt 2N/(N-2)$, $N \geq 3$, $\epsilon \gt 0$ is a parameter and $\lambda \in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. The potential $V$ is a bounded and continuous nonnegative function, satisfying some suitable global conditions. By employing the minimization techniques and the truncated argument, we obtain that the number of normalized solutions is not less than the number of global minimum points of $V$ when the parameter $\epsilon$ is sufficiently small.

Funding Statement

This research is supported by Guangdong Basic and Applied Basic Research Foundation (Nos. 2021A1515010383, 2022A1515010644), the Project of Science and Technology of Guangzhou (No. 202102020730).

Citation

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Lin Xu. Changxiu Song. Qilin Xie. "Multiplicity of Normalized Solutions for Schrödinger Equation with Mixed Nonlinearity." Taiwanese J. Math. 28 (3) 589 - 609, June, 2024. https://doi.org/10.11650/tjm/240202

Information

Received: 27 November 2023; Revised: 6 February 2024; Accepted: 20 February 2024; Published: June, 2024
First available in Project Euclid: 20 May 2024

Digital Object Identifier: 10.11650/tjm/240202

Subjects:
Primary: 35J20 , 35J60 , 47J30

Keywords: minimization , mixed nonlinearities , normalized solutions , truncated argument

Rights: Copyright © 2024 The Mathematical Society of the Republic of China

Vol.28 • No. 3 • June, 2024
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