单位球上Yamabe方程的全局分歧  

Global Bifurcation for the Yamabe Equation on the Unit Sphere

作　　者：代国伟[1] 高思雨 马如云 Dai Guowei;Gao Siyu;Ma Ruyun(School of Mathematical Sciences,Dalian University of Technology,Liaoning Dalian 116024;School of Mathematical and Statistics,Xidian University,Xi'an 710071)

机构地区：[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

出　　处：《数学物理学报（A辑）》2023年第5期1391-1396,共6页Acta Mathematica Scientia

基　　金：国家自然科学基金(11871129)。

摘　　要：该文研究了N维单位球面S^(N)上的Yamabe方程−Δ_(S^(N))v+λv=vN+2/N−2.通过分歧的方法,对于任意k≥1,证明了该方程对于任意的λ>λ_(k):=(k+N−1)(N−2)/4都至少有一个非常数解vk,使得vk−λ^(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≥4时,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.We study the Yamabe equation on the N-dimensional unit sphere S^(N)−Δ_(S^(N))v+λv=vN+2/N−2.By bifurcation technique,for each k≥1,we prove that this equation has at least one non-constant solution vk for anyλ>λ_(k):=(k+N−1)(N−2)/4 such that vk−λ^(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 R^(N) with n≥4.Moreover,we also obtain the global bifurcation results of the Yamabe problem in product manifolds with one of the manifold is the unit sphere.

关 键 词：分歧 Yamabe 方程 非径向解 

分 类 号：O177.91[理学—数学]