单位球上 Yamabe 方程的全局分歧

Abstract

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

CLC Number:

O177.91

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.

Trendmd

References 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

Global Bifurcation for the Yamabe Equation on the Unit Sphere

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References 20

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Chen Min,Hu Biyan,Luo Hong. Boundary Layer Separation of 2-D Incompressible Navier-Stokes-Allen-Cahn System [J]. Acta mathematica scientia,Series A, 2023, 43(4): 1123-1132.
[2]	Rui Xu,Yan Yang. Dynamics of an HTLV-I Infection Model with Delayed and Saturated CTL Immune Response and Immune Impairment [J]. Acta mathematica scientia,Series A, 2022, 42(6): 1836-1848.
[3]	Xueli Jiang,Xuan Deng,Qiuhao Wen,Yanqin Xiong. Limit Circle Bifurcations of Polynomial Differential System with Two Parallel Switch Straight Lines [J]. Acta mathematica scientia,Series A, 2022, 42(2): 353-364.
[4]	Daoxiang Zhang,Ben Li,Dandan Chen,Yating Lin,Xinmei Wang. Hopf Bifurcation for a Fractional Differential-Algebraic Predator-Prey System with Time Delay and Economic Profit [J]. Acta mathematica scientia,Series A, 2022, 42(2): 570-582.
[5]	Yue Sun,Daoxiang Zhang,Wen Zhou. The Influence of Fear Effect on Stability Interval of Reaction-Diffusion Predator-Prey System with Time Delay [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1980-1992.
[6]	Xin Xie,Jianquan Li,Yuping Wang,Dian Zhang. A Qualitative Analysis of a Tumor-Immune System with Antigenicity [J]. Acta mathematica scientia,Series A, 2021, 41(6): 1969-1979.
[7]	Bin Long,Shanshan Xu,Hui Cao,Jianquan Li. The Influence of Splitting Index on Heteroclinic Orbit Bifurcation Under Periodic Perturbation [J]. Acta mathematica scientia,Series A, 2021, 41(5): 1516-1528.
[8]	Jingnan Wang,Dezhong Yang. Stability and Bifurcation of a Pathogen-Immune Model with Delay and Diffusion Effects [J]. Acta mathematica scientia,Series A, 2021, 41(4): 1204-1217.
[9]	Xiaoyao Jia,Zhenluo Lou. Existence of Multiple Non-Radial Positive Solutions of a Hénon Type Elliptic System [J]. Acta mathematica scientia,Series A, 2021, 41(3): 723-728.
[10]	Leta Temesgen Desta,Wenjun Liu,Jian Ding. Dynamics of Traveling Wave Solutions to Fully Nonlinear Heavy Ion-Acoustic Degenerate Relativistic Quantum Plasmas [J]. Acta mathematica scientia,Series A, 2021, 41(2): 496-506.
[11]	Gaihui Guo,Xiaohui Liu. Hopf Bifurcation and Stability for an Autocatalytic Reversible Biochemical Reaction Model [J]. Acta mathematica scientia,Series A, 2021, 41(1): 166-177.
[12]	Mingmin Wang,Gao Jia. Bifurcation of Positive Solutions for Quasilinear Elliptic Equations with Φ-Laplacian Operator and Concave-Convex Nonlinearities [J]. Acta mathematica scientia,Series A, 2020, 40(5): 1235-1247.
[13]	Haixia Li. Existence and Uniqueness of Positive Solutions to an Unstirred Chemostat with Toxins [J]. Acta mathematica scientia,Series A, 2020, 40(5): 1175-1185.
[14]	Wang Lingjun, Wang Qiusi. Small-Amplitude Solitary Interfacial Traveling Waves in a Gravity-Capillary Two-Layered Fluid with Vorticity [J]. Acta mathematica scientia,Series A, 2020, 40(4): 947-976.
[15]	Yi Shao,Chunxiang A. Limit Cycles Bifurcations of Liénard System of Degree Four with One Nilpotent Cusp [J]. Acta mathematica scientia,Series A, 2020, 40(3): 619-630.