具有临界增长的分数阶 Kirchhoff 型方程半经典解的存在性、集中性和多解性

数学物理学报 ›› 2026, Vol. 46 ›› Issue (4): 1634-1666.

• • 上一篇

具有临界增长的分数阶 Kirchhoff 型方程半经典解的存在性、集中性和多解性

郭伦¹(), 黄文涛²(), 贾慧芳³(), 潘政¹^,^*()

¹ 中南民族大学数学与统计学学院武汉 430070
² 华东交通大学理学院南昌 330013
³ 广东工业大学数学与统计学院广州 510520

收稿日期:2026-04-05 修回日期:2026-05-20 出版日期:2026-08-26 发布日期:2026-06-10
通讯作者: 潘政 E-mail:lguo@mails.ccnu.edu.cn;wthuang1014@aliyun.com;hf_jia@mails.ccnu.edu.cn;2024110592@mail.scuec.edu.cn
作者简介:郭伦,E-mail: lguo@mails.ccnu.edu.cn;
黄文涛, E-mail: wthuang1014@aliyun.com;
贾慧芳,E-mail: hf_jia@mails.ccnu.edu.cn
基金资助:
湖北省自然科学基金(2024AFB839);江西省自然科学基金(20232BAB201009);广东省自然科学基金(2024A1515012370)

Existence, Concentration and Multiplicity of Semiclassical Solutions for a Fractional Kirchhoff Equation with Critical Growth

Lun Guo¹(), Wentao Huang²(), Huifang Jia³(), Zheng Pan¹^,^*()

¹ School of Mathematics and Statistics, South-Central Minzu University, Wuhan 430070
² School of Science, East China Jiaotong University, Nanchang 330013
³ School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou 510520

Received:2026-04-05 Revised:2026-05-20 Online:2026-08-26 Published:2026-06-10
Contact: Zheng Pan E-mail:lguo@mails.ccnu.edu.cn;wthuang1014@aliyun.com;hf_jia@mails.ccnu.edu.cn;2024110592@mail.scuec.edu.cn
Supported by:
Hubei Provincial Natural Science Foundation of China(2024AFB839);Natural Science Foundation of Jiangxi Province(20232BAB201009);Guangdong Basic and Applied Basic Research Foundation of China(2024A1515012370)

摘要/Abstract

摘要：

该文研究如下具有临界增长的分数阶 Kirchhoff 型方程

$ \left(\epsilon^{2s}a+\epsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}{\rm d}x\right) (-\Delta)^{s}u+V(x)u=K(x)f(u)+|u|^{2^{*}_{s}-2}u, \ \ u\in H^{s}(\mathbb{R}^{3}), $

其中 $\epsilon>0$ 为小参数, 常数 $a,b>0$, $s\in(\frac{3}{4},1)$, $2^ {*}_{s}=\frac{6}{3-2s}$ 是临界 Sobolev 指数, 势函数$ V,K:\mathbb{R}^{3}\to\mathbb{R}$ 是非负连续函数, $f:\mathbb{R} \to \mathbb{R}$ 为连续但不可微的次临界非线性项. 作者应用 [Szulkin A, Weth T. Boston: International Press, 2010] 提出的广义 Nehari 流形方法, 证明了基态解的存在性及其集中性质. 此外, 利用 Ljusternik-Schnirelmann 畴数理论, 建立方程解的个数与势函数 $V$ 达到其最小值的集合以及 $K$ 达到其最大值的集合的拓扑之间的关系.

关键词: 分数阶 Kirchhoff 型方程, 临界增长, 基态解, Ljusternik-Schnirelmann 畴数理论

Abstract:

This paper studies the following fractional Kirchhoff-type equation with critical growth

$ \left(\epsilon^{2s}a+\epsilon^{4s-3}b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}{\rm d}x\right) (-\Delta)^{s}u+V(x)u=K(x)f(u)+|u|^{2^{*}_{s}-2}u, \ \ u\in H^{s}(\mathbb{R}^{3}), $

where $\epsilon>0$ is a small parameter, $a,b>0$ are constants, $s\in(\frac{3}{4},1)$, $2^{*}_{s}=\frac{6}{3-2s}$ is the Sobolev critical exponent, the potential functions $V,K:\mathbb{R}^{3}\to\mathbb{R}$ are nonnegative continuous functions, and $f:\mathbb{R}\to\mathbb{R}$ is a continuous but non-differentiable subcritical nonlinear term. By using the generalized Nehari manifold method introduced by [Szulkin A, Weth T. Boston: International Press, 2010], the authors prove the existence of ground state solutions and their concentration properties. Furthermore, using Ljusternik-Schnirelmann category theory, they establish a relationship between the number of solutions and the topology of the sets where the potential $V$ attains its minimum and $K$ attains its maximum.

Key words: fractional Kirchhoff equation, critical growth, ground state solution, Ljusternik-Schnirelmann category theory

中图分类号:

O175.23

郭伦, 黄文涛, 贾慧芳, 潘政. 具有临界增长的分数阶 Kirchhoff 型方程半经典解的存在性、集中性和多解性[J]. 数学物理学报, 2026, 46(4): 1634-1666.

Lun Guo, Wentao Huang, Huifang Jia, Zheng Pan. Existence, Concentration and Multiplicity of Semiclassical Solutions for a Fractional Kirchhoff Equation with Critical Growth[J]. Acta mathematica scientia,Series A, 2026, 46(4): 1634-1666.

参考文献 46

[1]	Alves C O, Correa F J S A. On existence of solutions for a class of problem involving a nonlinear operator. Comm Appl Nonlinear Anal, 2001, 8(2): 43-56
[2]	Alves C O, Miyagaki O H. Existence and concentration of solution for a class of fractional elliptic equation in $\mathbb{R}^{N}$ via penalization method. Calc Var Partial Differential Equations, 2016, 55(3): Art 47
[3]	Ambrosio V, Isernia T. Concentration phenomena for a fractional Schrödinger-Kirchhoff type equation. Math Methods Appl Sci, 2018, 41(2): 615-645 doi: 10.1002/mma.v41.2
[4]	Arosio A, Panizzi S. On the well-posedness of the Kirchhoff string. Trans Amer Math Soc, 1996, 348(1): 305-330 doi: 10.1090/tran/1996-348-01
[5]	Autuori G, Fiscella A, Pucci P. Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal, 2015, 125: 699-714 doi: 10.1016/j.na.2015.06.014
[6]	Benci V, Cerami G. Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology. Calc Var Partial Differential Equations, 1994, 2(1): 29-48 doi: 10.1007/BF01234314
[7]	Caffarelli L, Silvestre L. An extension problem related to the fractional Laplacian. Commun Partial Differential Equations, 2007, 32(8): 1245-1260 doi: 10.1080/03605300600987306
[8]	Chang K C. Infinite Dimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Differential Equations and Their Applications. Boston: Birkhauser Boston Inc, 1993
[9]	Chipot M, Lovat B. Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal, 1997, 30(7): 4619-4627 doi: 10.1016/S0362-546X(97)00169-7
[10]	D'Ancona P, Spagnolo S. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent Math, 1992, 108(1): 247-262 doi: 10.1007/BF02100605
[11]	D'avila J, Del Pino M, Wei J. Concentrating standing waves for the fractional nonlinear Schrödinger equation. J Differential Equations, 2014, 256(2): 858-892 doi: 10.1016/j.jde.2013.10.006
[12]	Del Pino M, Felmer P L. Local mountain pass for semilinear elliptic problems in unbounded domains. Calc Var Partial Differential Equations, 1996, 4(2): 121-137 doi: 10.1007/BF01189950
[13]	Dipierro S, Medina M, Valdinoci E. Fractional Elliptic Problems with Critical Growth in the Whole of $\mathbb{R}^{N}$. Pisa: Edizioni della Normale Pisa, 2017
[14]	Fall M, Mahmoudi F, Valdinoci E. Ground states and concentration phenomena for the fractional Schrödinger equation. Nonlinearity, 2015, 28: 1937-1961 doi: 10.1088/0951-7715/28/6/1937
[15]	Felmer P, Quass A, Tan J. Positive solutions of the nonlinear Schrödinger equation with the fractional Laplacian. Proc Roy Soc Edinburgh Sect A, 2012, 142: 1237-1262 doi: 10.1017/S0308210511000746
[16]	Fiscella A, Valdinoci E. A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal, 2014, 94: 156-170 doi: 10.1016/j.na.2013.08.011
[17]	Ghoussoub N. Duality and Perturbation Methods in Critical Point Theory. Cambridge: Cambridge University Press, 1993
[18]	Guo Q, He X. Semiclassical states for fractional Schrödinger equations with critical growth. Nonlinear Anal, 2017, 151: 164-186 doi: 10.1016/j.na.2016.12.004
[19]	He X, Zou W. Ground states for nonlinear Kirchhoff equations with critical growth. Ann Mat Pura Appl, 2014, 193: 473-500 doi: 10.1007/s10231-012-0286-6
[20]	He X, Zou W. Existence and concentration result for the fractional Schrödinger equations with critical nonlinearities. Calc Var Partial Differential Equations, 2016, 55(4): Art 91
[21]	He X, Zou W. Multiplicity of concentrating solutions for a class of fractional Kirchhoff equation. Manuscripta Math, 2019, 158(1): 159-203 doi: 10.1007/s00229-018-1017-0
[22]	He Y, Li G. Standing waves for a class of Kirchhoff type problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents. Calc Var Partial Differential Equations, 2015, 54(3): 3067-3106 doi: 10.1007/s00526-015-0894-2
[23]	He Y, Li G, Peng S. Concentrating bound states for Kirchhoff type problems in $\mathbb{R}^{3}$ involving critical Sobolev exponents. Adv Nonlinear Stud, 2014, 14: 483-510 doi: 10.1515/ans-2014-0214
[24]	Kirchhoff G. Mechanik. Leipzig: Teubner, 1883
[25]	Laskin N. Fractional quantum mechanics and Levy path integrals. Phys Lett A, 2000, 268(4/6): 298-305 doi: 10.1016/S0375-9601(00)00201-2
[26]	Laskin N. Fractional Schrödinger equation. Phys Rev E (3), 2002, 66(5): Art 056108
[27]	Li G, Ye H. Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in $\mathbb{R}^{3}$. J Differential Equations, 2014, 257(2): 566-600 doi: 10.1016/j.jde.2014.04.011
[28]	Li S, Ding Y, Chen Y. Concentrating standing waves for the fractional Schrödinger equation with critical nonlinearities. Bound Value Probl, 2015, 2015(1): Art 240
[29]	Liang S, Repovs D, Zhang B. On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity. Comput Math Appl, 2018, 75(5): 1778-1794 doi: 10.1016/j.camwa.2017.11.033
[30]	Lions J L. On some questions in boundary value problems of mathematical physics in contemporary developments in continuum mechanics and partial differential equations. North-Holl Math Stud, 1978, 30: 284-346
[31]	Lions P L. The concentration-compactness principle in the calculus of variations, the limit case I. Rev Mat Iberoam, 1985, 1: 145-201 doi: 10.4171/rmi
[32]	Lions P L. The concentration-compactness principle in the calculus of variations, the limit case II. Rev Mat Iberoam, 1985, 1: 45-121
[33]	Liu Z, Guo S. Existence and concentration of positive ground states for a Kirchhoff equation involving critical Sobolev exponent. Z Angew Math Phys, 2015, 66(3): 747-769 doi: 10.1007/s00033-014-0431-8
[34]	Liu Z, Squassina M, Zhang J. Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension. Nonlinear Differential Equations Appl, 2017, 24(4): Art 50
[35]	Nezza E D, Palatucci G, Valdinoci E. Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136(5): 521-573 doi: 10.1016/j.bulsci.2011.12.004
[36]	Palatucci G, Pisante A. Improved Sobolev embeddings, profile decomposition, and concentration-compactness for fractional Sobolev spaces. Calc Var Partial Differential Equations, 2014, 50(3): 799-829 doi: 10.1007/s00526-013-0656-y
[37]	Pucci P, Saldi S. Critical stationary Kirchhoff equations in $\mathbb{R}^{N}$ involving nonlocal operators. Rev Mat Iberoam, 2016, 32: 1-22 doi: 10.4171/rmi
[38]	Secchi S. Ground state solutions for nonlinear fractional Schrödinger equations in $\mathbb{R}^{N}$. J Math Phys, 2013, 54(3): Art 031501
[39]	Servadei R, Valdinoci E. The Brezis-Nirenberg result for the fractional Laplacian. Trans Amer Math Soc, 2015, 367(1): 67-102 doi: 10.1090/tran/2015-367-01
[40]	Shang X, Zhang J. Ground states for fractional Schrödinger equations with critical growth. Nonlinearity, 2014, 27: 187-207 doi: 10.1088/0951-7715/27/2/187
[41]	Silvestre L. Regularity of the obstacle problem for a fractional power of the Laplace operator. Commun Pure Appl Math, 2007, 60: 67-112 doi: 10.1002/cpa.v60:1
[42]	Szulkin A, Weth T. The method of Nehari manifold//Gao D Y, Motreanu D. Handbook of Nonconvex Analysis and Applications. Somerville: International Press, 2010: 597-632.
[43]	Wang J, Xiao L. Existence and concentration of solutions for a Kirchhoff type problem with potentials. Discrete Contin Dyn Syst, 2016, 36: 7137-7168 doi: 10.3934/dcds.2016111
[44]	Willem M. Minimax Theorems. Boston: Birkhauser Boston Inc, 1996.
[45]	Xiang M, Zhang B, Qiu H. Existence of solutions for a critical fractional Kirchhoff type problem in $\mathbb{R}^{N}$. Sci China Math, 2017, 60(9): 1647-1660 doi: 10.1007/s11425-015-0792-2
[46]	Zhang X, Zhang C. Existence of solutions for critical fractional Kirchhoff problems. Math Methods Appl Sci, 2017, 40(5): 1649-1665 doi: 10.1002/mma.v40.5

具有临界增长的分数阶 Kirchhoff 型方程半经典解的存在性、集中性和多解性

Existence, Concentration and Multiplicity of Semiclassical Solutions for a Fractional Kirchhoff Equation with Critical Growth

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