单位球上 Yamabe 方程的全局分歧

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 1391-1396.

单位球上 Yamabe 方程的全局分歧

代国伟^1,^*(),高思雨¹,马如云²

¹大连理工大学数学科学学院辽宁大连 116024
²西安电子科技大学数学与统计学院西安 710071

收稿日期:2022-08-14 修回日期:2023-03-23 出版日期:2023-10-26 发布日期:2023-08-09
通讯作者: 代国伟 E-mail:daiguowei@dlut.edu.cn
基金资助:
国家自然科学基金(11871129)

Global Bifurcation for the Yamabe Equation on the Unit Sphere

Dai Guowei^1,^*(),Gao Siyu¹,Ma Ruyun²

¹School of Mathematical Sciences, Dalian University of Technology, Liaoning Dalian 116024
²School of Mathematical and Statistics, Xidian University, Xi'an 710071

Received:2022-08-14 Revised:2023-03-23 Online:2023-10-26 Published:2023-08-09
Contact: Guowei Dai E-mail:daiguowei@dlut.edu.cn
Supported by:
NSFC(11871129)

摘要/Abstract

摘要：

该文研究了 $N$ 维单位球面 $\mathbb{S}^N$ 上的Yamabe方程

$\begin{equation} -\Delta_{\mathbb{S}^N} v+\lambda v=v^{\frac{N+2}{N-2}}.\nonumber \end{equation}$

通过分歧的方法, 对于任意 $k\geq1$ , 证明了该方程对于任意的 $\lambda>\lambda_k:=(k+N-1)(N-2)/4$ 都至少有一个非常数解 $v_k$ , 使得 $v_k-\lambda^{1/(N^{*}-1)}$ 正好有 $k$ 个零点, 并且它们在 $(-1, 1)$ 中都是单根, 其中 $N^{*}$ 是 Sobolev 临界指数. 在应用部分, 得到了当 $n\geq4$ 时, $\mathbb{R}^N$ 上非线性椭圆方程非径向解的存在性. 此外, 还得到了乘积流形中一个流形是单位球时的 Yamabe 问题的全局分歧结果.

关键词: 分歧, Yamabe 方程, 非径向解

Abstract:

We study the Yamabe equation on the $N$ -dimensional unit sphere $\mathbb{S}^N$

$\begin{equation} -\Delta_{\mathbb{S}^N} v+\lambda v=v^{\frac{N+2}{N-2}}.\nonumber \end{equation}$

By bifurcation technique, for each $k\geq1$ , we prove that this equation has at least one non-constant solution $v_k$ for any $\lambda>\lambda_k:=(k+N-1)(N-2)/4$ such that $v_k-\lambda^{1/(N^{*}-1)}$ has exactly $k$ zeroes, all of them are in $(-1, 1)$ and are simple, where $N^{*}$ is the sobolev critical exponent. As application, we obtain the existence of non-radial solutions of a nonlinear elliptic equation on $\mathbb{R}^N$ with $n\geq4$ . Moreover, we also obtain the global bifurcation results of the Yamabe problem in product manifolds with one of the manifold is the unit sphere.

Key words: Bifurcation, Yamabe equation, Non-radial solutions

中图分类号:

O177.91

代国伟,高思雨,马如云. 单位球上 Yamabe 方程的全局分歧[J]. 数学物理学报, 2023, 43(5): 1391-1396.

Dai Guowei,Gao Siyu,Ma Ruyun. Global Bifurcation for the Yamabe Equation on the Unit Sphere[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1391-1396.

参考文献 20

[1]	Aubin T. Onlinear Analysis on Manifolds. Monge-Ampère Equations. New York: Springer-Verlag, 1982
[2]	Bettiol R, Piccione P. Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions. Pacific J Math, 2013, 266: 1-21 doi: 10.2140/pjm
[3]	Bettiol R, Piccione P. Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres. Calc Var Partial Differential Equations, 2013, 47: 789-807 doi: 10.1007/s00526-012-0535-y
[4]	Bérard-Bergery L. La courbure scalaire de variétés riemanniennes//Séminaire Bourbaki. Berlin, Heidelberg: Springer, 2006: 225-245
[5]	Brendle S. Blow-up phenomena for the Yamabe equation. J Amer Math Soc, 2008, 21: 951-979 doi: 10.1090/jams/2008-21-04
[6]	Caffarelli L A, Gidas B, Spruck J. Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm Pure Appl Math, 1989, 42: 271-297 doi: 10.1002/(ISSN)1097-0312
[7]	Dai G. Two Whyburn type topological theorems and its applications to Monge-Ampère equations. Calc Var Partial Differential Equations, 2016, 55: 97 doi: 10.1007/s00526-016-1029-0
[8]	Dai G. Bifurcation and one-sign solutions of the $p$ -Laplacian involving a nonlinearity with zeros. Discrete Contin Dyn Syst, 2016, 36: 5323-5345 doi: 10.3934/dcdsa
[9]	de Lima L L, Piccione P, Zedda M. On bifurcation of solutions of the Yamabe problem in product manifolds. Ann Inst H Poincaré Anal Non Linéaire, 2012, 29: 261-277 doi: 10.4171/aihpc
[10]	Gidas B, Ni W M, Nirenberg L. Symmetry and related properties via the maximum principle. Comm Math Phys, 1979, 68: 209-243 doi: 10.1007/BF01221125
[11]	Henry G, Petean J. Isoparametric hypersurfaces and metrics of constant scalar curvature. Asian J Math, 2014, 18: 53-68 doi: 10.4310/AJM.2014.v18.n1.a3
[12]	Jin Q, Li Y, Xu H. Symmetry and asymmetry: The method of moving spheres. Adv Differential Equations, 2008, 13: 601-640
[13]	Kobayashi O. Scalar curvature of a metric with unit volume. Math Ann, 1987, 279: 253-265 doi: 10.1007/BF01461722
[14]	Obata M. The conjecture on conformal transformations of Riemannian manifolds. J Diff Geom, 1971, 6: 247-258
[15]	Petean J. Metrics of constant scalar curvature conformal to Riemannian products. Proc Amer Math Soc, 2010, 138: 2897-2905 doi: 10.1090/S0002-9939-10-10293-7
[16]	Petean J. Multiplicity results for the Yamabe equation by Lusternik-Schnirelmann theory. J Funct Anal, 2019, 276: 1788-1805 doi: 10.1016/j.jfa.2018.08.011
[17]	Pollack D. Nonuniqueness and high energy solutions for a conformally invariant scalar equation. Comm Anal Geom, 1993, 1: 347-414 doi: 10.4310/CAG.1993.v1.n3.a2
[18]	Schoen R. Variational theory for the total scalar curvature functional for Riemannian metrics and related topics//Lecture Notes in Math, Vol 1365. Berlin: Springer-Verlag, 1989: 120-154
[19]	Whyburn G T. Topological Analysis. Princeton: Princeton University Press, 1958
[20]	Yamabe H. On the deformation of Riemannian structures on compact manifolds. Osaka Math J, 1960, 12: 21-37

Metrics

Viewed

Full text

279

HTML			PDF

Just accepted	Online first	Issue	Just accepted	Online first	Issue
0	0	3	0	0	276

	From	local

	Times	279
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	93

From	Others	local

Times	88	5
Rate	95%	5%

Cited

Web of Science	Crossref	ScienceDirect	Search for Citations in Google Scholar >>


This page requires you have already subscribed to WoS.

Shared

Discussed