POHOZAEV MINIMIZERS FOR FRACTIONAL CHOQUARD EQUATIONS WITH MASS-SUPERCRITICAL NONLINEARITY

doi:10.1007/s10473-026-0111-8

数学物理学报(英文版) ›› 2026, Vol. 46 ›› Issue (1): 164-188.doi: 10.1007/s10473-026-0111-8

POHOZAEV MINIMIZERS FOR FRACTIONAL CHOQUARD EQUATIONS WITH MASS-SUPERCRITICAL NONLINEARITY

Liju WU¹, Jiankang XIA^2,3,*

1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China;
2. Shenzhen Research Institute of Northwestern Polytechnical University, Shenzhen 518057, China;
3. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China

收稿日期:2024-09-02 修回日期:2025-03-11 出版日期:2026-01-25 发布日期:2026-05-22

POHOZAEV MINIMIZERS FOR FRACTIONAL CHOQUARD EQUATIONS WITH MASS-SUPERCRITICAL NONLINEARITY

Liju WU¹, Jiankang XIA^2,3,*

1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China;
2. Shenzhen Research Institute of Northwestern Polytechnical University, Shenzhen 518057, China;
3. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, China

Received:2024-09-02 Revised:2025-03-11 Online:2026-01-25 Published:2026-05-22
Contact: * Jiankang Xia, E-mail: jiankangxia@nwpu.edu.cn
About author:Liju Wu, E-mail: 15035381897@163.com
Supported by:
Guangdong Basic and Applied Basic Research Foundation (2022A1515012138) and the NSFC (12271436, 12371119). J. Xia was supported by the Natural Science Basic Research Program of Shaanxi (2022JC-04).

摘要/Abstract

摘要： We investigate the constrained fractional Choquard equation
$\begin{align*} \begin{cases} (-\Delta)^s u=(I_{\alpha}*F(u))F'(u)-\mu u,\; \text{ in } \, \mathbb R^N,\\ u\in H^s(\mathbb R^N),\\ \displaystyle \int_{\mathbb R^N}|u|^2\mathrm{d} x=m, \end{cases} \end{align*}$
where $m>0$, $N> 2s $ with $s\in(0,1)$ being the order of the fractional Laplacian operator and $I_\alpha$ for $\alpha\in(0,N)$ denotes the Riesz potential. The parameter $\mu\in\mathbb R$ appears as a Lagrange multiplier. By imposing general mass-supercritical conditions on $F$, we confirm the existence of normalized solutions that characterize the global minimizer on the Pohozaev manifold. Our proof does not depend on the assumption that all weak solutions satisfy the Pohozaev identity, a challenge that remains unsolved for this doubly nonlocal equation.

关键词: nonlinear fractional Choquard equation, double nonlocality, super-critical mass, normalized solutions, Pohozaev minimizer

Abstract: We investigate the constrained fractional Choquard equation
$\begin{align*} \begin{cases} (-\Delta)^s u=(I_{\alpha}*F(u))F'(u)-\mu u,\; \text{ in } \, \mathbb R^N,\\ u\in H^s(\mathbb R^N),\\ \displaystyle \int_{\mathbb R^N}|u|^2\mathrm{d} x=m, \end{cases} \end{align*}$
where $m>0$, $N> 2s $ with $s\in(0,1)$ being the order of the fractional Laplacian operator and $I_\alpha$ for $\alpha\in(0,N)$ denotes the Riesz potential. The parameter $\mu\in\mathbb R$ appears as a Lagrange multiplier. By imposing general mass-supercritical conditions on $F$, we confirm the existence of normalized solutions that characterize the global minimizer on the Pohozaev manifold. Our proof does not depend on the assumption that all weak solutions satisfy the Pohozaev identity, a challenge that remains unsolved for this doubly nonlocal equation.

Key words: nonlinear fractional Choquard equation, double nonlocality, super-critical mass, normalized solutions, Pohozaev minimizer

中图分类号:

35A01

Liju WU¹, Jiankang XIA^2,3,*. POHOZAEV MINIMIZERS FOR FRACTIONAL CHOQUARD EQUATIONS WITH MASS-SUPERCRITICAL NONLINEARITY[J]. 数学物理学报(英文版), 2026, 46(1): 164-188.

Liju WU¹, Jiankang XIA^2,3,*. POHOZAEV MINIMIZERS FOR FRACTIONAL CHOQUARD EQUATIONS WITH MASS-SUPERCRITICAL NONLINEARITY[J]. Acta mathematica scientia,Series B, 2026, 46(1): 164-188.

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POHOZAEV MINIMIZERS FOR FRACTIONAL CHOQUARD EQUATIONS WITH MASS-SUPERCRITICAL NONLINEARITY

POHOZAEV MINIMIZERS FOR FRACTIONAL CHOQUARD EQUATIONS WITH MASS-SUPERCRITICAL NONLINEARITY

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