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.

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