分数阶Kirchhoff型约束变分问题Schwarz对称极小解的存在性

数学物理学报 ›› 2022, Vol. 42 ›› Issue (6): 1719-1728.

分数阶Kirchhoff型约束变分问题Schwarz对称极小解的存在性

魏均平(),黄小梦(),张贻民*()

^{武汉理工大学数学科学研究中心, 武汉 430070}

收稿日期:2022-01-07 出版日期:2022-12-26 发布日期:2022-12-16
通讯作者: 张贻民 E-mail:weijunpingacca@163.com;hhuangxiaomeng@126.com;zhangym802@126.com
作者简介:魏均平, E-mail: weijunpingacca@163.com|黄小梦, E-mail: hhuangxiaomeng@126.com
基金资助:
国家自然科学基金(12171379)

Existence of Schwarz Symmetric Minimizers for Fractional Kirchhoff Constrained Variational Problem

Junping Wei(),Xiaomeng Huang(),Yimin Zhang*()

^{Center for Mathematical Sciences, Wuhan University of Technology, Wuhan 430070}

Received:2022-01-07 Online:2022-12-26 Published:2022-12-16
Contact: Yimin Zhang E-mail:weijunpingacca@163.com;hhuangxiaomeng@126.com;zhangym802@126.com
Supported by:
the NSFC(12171379)

摘要/Abstract

摘要：

该文主要讨论一般非线性项情形下一类分数阶Kirchhoff型约束变分问题在空间$H^s(\mathbb{R}^N)$上极小解的存在性和Schwarz对称性.利用单调递减重排不等式和伸缩技巧, 当非线性项满足合适的条件时, 该文细致的讨论了相应约束变分问题在空间$H^s(\mathbb{R}^N)$上极小解的存在性与非线性项的指数和约束$\int_{\mathbb{R}^N}|u|^2{\rm d}x=c^2$中参数$c$的关系.

关键词: 约束变分, 极小解, 分数阶Kirchhoff问题, Schwarz对称

Abstract:

In this paper, the existence and Schwarz symmetry of minimizers for a fractional Kirchhoff constrained variational problem with general nonlinear term were studied in space $H^s(\mathbb{R} ^N)$. Using symmetric decreasing rearrangement inequality and scaling technique, the relation between existence of minimizers for a fractional Kirchhoff constrained variational problem with the exponent of nonlinear term and parameter $c$ in $\int_{\mathbb{R} ^N}|u|^2{\rm d}x=c^2$ was discussed.

Key words: Constrained variational, Minimizer, Fractional Kirchhoff problem, Schwarz symmetric

中图分类号:

O175.25

魏均平,黄小梦,张贻民. 分数阶Kirchhoff型约束变分问题Schwarz对称极小解的存在性[J]. 数学物理学报, 2022, 42(6): 1719-1728.

Junping Wei,Xiaomeng Huang,Yimin Zhang. Existence of Schwarz Symmetric Minimizers for Fractional Kirchhoff Constrained Variational Problem[J]. Acta mathematica scientia,Series A, 2022, 42(6): 1719-1728.

参考文献

1	Amick C J , Toland J F . Uniqueness and related analytic properties for the Benjamin-Ono equation-a nonlinear Neumann problem in the plane. Acta Mathematica, 1991, 167 (1): 107- 126
2	Chen W X , Li C M , Ou B . Classification of solutions for an integral equation. Communications on Pure and Applied Mathematics, 2006, 59 (3): 330- 343
3	Nezza E Di , Palatucci G , Valdinoci E . Hitchhiker's guide to the fractional Sobolev spaces. Bull Sci Math, 2012, 136 (5): 521- 573
4	Ou B , Li C , Chen W . Qualitative properties of solutions for an integral equation. Discrete and Continuous Dynamical Systems-Series A, 2017, 12 (2): 347- 354
5	Li Y Y . Remark on some conformally invariant integral equations: the method of moving spheres. Journal of the European Mathematical Society, 2004, 6 (2): 153- 180
6	Frank R L , Lenzmann E . Uniqueness of non-linear ground states for fractional Laplacians in $\mathbb{R}^{N}$. Acta Mathematica, 2013, 210 (2): 261- 318
7	Frank R L , Lenzmann E , Silvestre L . Uniqueness of radial solutions for the fractional Laplacian. Communications on Pure and Applied Mathematics, 2013, 69 (9): 1671- 1726
8	成艺群, 滕凯民. 非线性临界Kirchhoff型问题的正基态解. 数学物理学报, 2021, 41A (3): 666- 685
8	Cheng Y Q , Teng K M . Positive ground state solutions for nonlinear critical Kirchhoff type problem. Acta Mathematica Scientia, 2021, 41A (3): 666- 685
9	Deng Y B , Peng S J , Shuai W . Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in $\mathbb{R}^{3}$. J Funct Anal, 2015, 269 (11): 3500- 3527
10	郭合林, 王云波. 关于一个约束变分问题的注记. 数学物理学报, 2017, 37A (6): 1125- 1128
10	Guo H L , Wang Y B . A remark on a constrained variational problem. Acta Mathematica Scientia, 2017, 37A (6): 1125- 1128
11	Ye H Y . The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations. Math Methods Appl Sci, 2015, 38 (13): 2663- 2679
12	柳志德, 王征平. 非线性Kirchhoff型椭圆方程的最低能量解. 数学物理学报, 2019, 39A (2): 264- 276
12	Liu Z D , Wang Z P . Least energy solution for nonlinear Kirchhoff type elliptic equation. Acta Mathematica Scientia, 2019, 39A (2): 264- 276
13	魏美春, 唐春雷. $\mathbb{R}^{N}$上的Kirchhoff型问题非平凡解的存在性和多解性. 数学物理学报, 2015, 35A (1): 151- 162
13	Wei M C , Tang C L . Existence and multiplicity of nontrivial solutions for Kirchhoff-type problem in begin{document}$mathbb{R}^{N}$end{document}. Acta Mathematica Scientia, 2015, 35A (1): 151- 162
14	梁文翠, 张正杰. Kirchhoff型方程有关的非线性方程多解的存在性. 数学物理学报, 2020, 40A (4): 842- 849
14	Liang W C , Zhang Z J . Multiple solutions for nonlinear equations related to Kirchhoff type equations. Acta Mathematica Scientia, 2020, 40A (4): 842- 849
15	Mao A M , Zhang Z T . Sign-changing and multiple solutions of Kirchhoff type problems without P.S. condition. Nonlinear Anal, 2009, 70 (3): 1275- 1287
16	He X M , Zou W M . Existence and concentration behavior of positive solutions for a Kirchhoff equation in $\mathbb{R}^{3}$. J Differ Equ, 2012, 252 (2): 1813- 1834
17	Li G B , Niu Y H . The existence and local uniqueness of multi-peak positive solutions to a class of Kirchhoff equation. Acta Mathematica Scientia, 2020, 40B (1): 90- 112
18	许诗敏, 王春花. Kirchhoff方程单峰解的局部唯一性. 数学物理学报, 2020, 40A (2): 432- 440
18	Xu S M , Wang C H . Local uniqueness of a single peak solution of a subcritical Kirchhoff problem in begin{document}$\mathbb{R}^{3}$end{document}. Acta Mathematica Scientia, 2020, 40A (2): 432- 440
19	Zeng X Y , Zhang Y M . Existence and uniqueness of normalized solutions for the Kirchhoff equation. Applied Math Letters, 2017, 74, 52- 59
20	Guo H L , Zhang Y M , Zhou H S . Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Commun Pure Appl Aanl, 2018, 17 (5): 1875- 1897
21	李容星, 王文清, 曾小雨. 带椭球势阱的Kirchhoff型方程的变分问题. 数学物理学报, 2019, 39A (6): 1323- 1333
21	Li R X , Wang W Q , Zeng X Y . A constrained variational problem of Kirchhoff type equation with ellipsoid-shaped potential. Acta Mathematica Scientia, 2019, 39A (6): 1323- 1333
22	Liu Z S, Squassina M, Zhang J J. Ground states for fractional Kirchhoff equations with critical nonlinearity in low dimension. Nonlinear Differential Equations and Applications, 2017, 24: Article number 50
23	Cheng K , Gao Q . Sign-changing solutions for the stationary Kirchhoff problems involving the fractional Laplacian in $\mathbb{R}^{N}$. Acta Mathematica Scientia, 2018, 38B (6): 1712- 1730
24	Pucci P , Xiang M Q , Zhang B L . Multiple solutions for nonhomogeneous Schr?dinger-Kirchhoff equations involving the fractional p-Laplacian in $\mathbb{R}^{N}$. Calc Var Partial Differ Equ, 2015, 54, 2785- 2806
25	Huang X M , Zhang Y M . Existence and uniqueness of minimizers for L²-constrained problems related to fractional Kirchhoff equation. Mathematical Methods in the Applied Sciences, 2020, 43 (15): 8763- 8775
26	Frank R L , Seiringer R . Non-linear ground state representations and sharp Hardy inequalities. J Fun Anal, 2008, 255 (12): 3407- 3430
27	Burchard A , Hajaiej H . Rearrangement inequalities for functionals with monotone integrands. Journal of Functional Analysis, 2006, 233 (2): 561- 582

[1]	李仁华, 王征平. 含强制位势的分数阶薛定谔泊松方程的正规化解[J]. 数学物理学报, 2023, 43(6): 1723-1730.
[2]	熊晨, 高琦. 一类耦合Ginzburg-Landau系统的局部极小解[J]. 数学物理学报, 2023, 43(2): 321-340.
[3]	杨迎,沈烈军. Chern-Simons-Schrödinger方程能量泛函的L²约束极小化问题[J]. 数学物理学报, 2022, 42(3): 716-729.
[4]	杨连峰,曾小雨. 拟相对论薛定谔方程基态解的存在性与爆破行为[J]. 数学物理学报, 2022, 42(1): 165-175.
[5]	李振杰,李磊. 带有凸非线性项的分数阶Laplace方程的解的对称性[J]. 数学物理学报, 2020, 40(5): 1224-1234.
[6]	李容星,王文清,曾小雨. 带椭球势阱的Kirchhoff型方程的变分问题[J]. 数学物理学报, 2019, 39(6): 1323-1333.
[7]	郭合林, 王云波. 关于一个约束变分问题的注记[J]. 数学物理学报, 2017, 37(6): 1125-1128.
[8]	谷龙江, 孙志禹, 曾小雨. 一类约束变分问题极小元的存在性及其集中行为[J]. 数学物理学报, 2017, 37(3): 510-518.
[9]	李静, 陈才生. R^N上一类拟线性Schrödinger方程的Nehari流形解[J]. 数学物理学报, 2016, 36(3): 507-520.
[10]	朱文兴. 总体优化一类双参数填充函数算法的改进[J]. 数学物理学报, 1999, 19(5): 550-558.