$p$-Laplacian 方程的 $L^2$ 约束问题: 质量超临界情况

数学物理学报 ›› 2026, Vol. 46 ›› Issue (1): 1-30.

• 研究论文 • 下一篇

$p$-Laplacian 方程的 $L^2$ 约束问题: 质量超临界情况

田雨陆^*(), 王灯山(), 赵亮()

北京师范大学数学科学学院教育部数学与复杂系统重点实验室北京 100875

收稿日期:2024-05-22 修回日期:2025-05-01 出版日期:2026-02-26 发布日期:2026-01-19
通讯作者: 田雨陆 E-mail:tianyl@mail.bnu.edu.cn;dswang@bnu.edu.cn;liangzhao@bnu.edu.cn
作者简介:王灯山, Email: dswang@bnu.edu.cn;
赵亮, Email: liangzhao@bnu.edu.cn
基金资助:
国家自然科学基金(12271039);国家自然科学基金(12371247);北京师范大学教育部数学与复杂系统重点实验室开放课题(K202303)

Normalized Solutions to a $p$-Laplacian Equation with an $L^2$ Constraint: The Mass Supercritical Case

Yulu Tian^*(), Dengshan Wang(), Liang Zhao()

School of Mathematical Sciences Key Laboratory of Mathematics and Complex Systems of MOE, Beijing Normal University, Beijing 100875

Received:2024-05-22 Revised:2025-05-01 Online:2026-02-26 Published:2026-01-19
Contact: Yulu Tian E-mail:tianyl@mail.bnu.edu.cn;dswang@bnu.edu.cn;liangzhao@bnu.edu.cn
Supported by:
NSFC(12271039);NSFC(12371247);Open Project Program of Key Laboratory of Mathematics and Complex Systems, Beijing Normal University(K202303)

摘要/Abstract

摘要：

In this paper, we study the existence of ground state solutions to the following $p$-Laplacian equation with an $L^2$ constraint

$$\begin{equation*} \begin{cases} -\Delta_{p}u+{\vert u\vert}^{p-2}u=f(u)-\mu u, x\in\mathbb{R}^N,\\ {\Vert u\Vert}^2_{L^2(\mathbb{R}^N)}=m,\\ u\in W^{1,p}(\mathbb{R}^N)\cap L^2(\mathbb{R}^N), \end{cases} \end{equation*}$$

where $N\geq3$, $2\leq p<N$, $m>0$, the nonlinearity $f\in C(\mathbb{R},\mathbb{R})$ satisfies the mass supercritical conditions and $\mu\in\mathbb{R}$ appears as a Lagrange multiplier. We also reduce the conditions for the nonlinearity $f$ and analyse the behavior of the ground state energy $E_m$ for $m>0$ which generalize the results for the nonlinear scalar field equation with $p=2$.

关键词: $p$-Laplacian 方程, 归一化解, 基态解, 质量约束

Key words: $p$-Laplacian equation, normalized solution, ground state energy, mass constraint

中图分类号:

O175.29

田雨陆, 王灯山, 赵亮. $p$-Laplacian 方程的 $L^2$ 约束问题: 质量超临界情况[J]. 数学物理学报, 2026, 46(1): 1-30.

Yulu Tian, Dengshan Wang, Liang Zhao. Normalized Solutions to a $p$-Laplacian Equation with an $L^2$ Constraint: The Mass Supercritical Case[J]. Acta mathematica scientia,Series A, 2026, 46(1): 1-30.

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$p$-Laplacian 方程的 $L^2$ 约束问题: 质量超临界情况

Normalized Solutions to a $p$-Laplacian Equation with an $L^2$ Constraint: The Mass Supercritical Case

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