论文标题
Smace Yamabe问题的ode降低方法
An ODE reduction method for the semi-Riemannian Yamabe problem on space forms
论文作者
论文摘要
我们考虑该形式的半摩曼山山型方程 \ [ - \ square u +λu=μ\ vert u \ vert^{p-1} u \ quad \ text {on} m \]其中$ m $是$ m \ m \ geq 3 $，$ \ square $的半欧糖层空间或伪层，是$ m $，$ m $，$λ\ geq0 $，$ mathbb in \ MathBB {r} \ r} \ smalleptemsetminus \ $ pp> $ pp> $ pp> $ pp> $ pp> $ pp> $ pp> $ pp> $ p> $ pp> $ pp> $ pp> $ pp> $ pp> $ pp> $ pp> $ pp> $ pp）使用$ m $上的半摩恩尼亚等式等级功能，我们将pde降低到形式的广义emden-fowler ode中\ [w'''+q（r）w'+λw=μ\ vert w \ vert w \ vert w \ vert w \ vert^{p-1} {p-1} $ [0，\ infty）$或$ [0，π] $，$ q（r）$以$ 0 $ blup-up up $ 0 $，$ w $受自然初始条件的约束$ w'（0）= 0 $在第一种情况下，$ w'（0）= w'（π）= 0 $。我们证明了对该问题的爆炸和全球定义的解决方案的存在，无论是积极的还是改变了签名，诱导了具有相同定性特性的半利马尼亚尼亚Yamabe型问题，其水平和关键集都在半利马尼亚异端层状溶液和局灶性层次表面和局灶性杂种中描述。特别是，我们证明了具有规定数量的淋巴结域数量的伪层中半摩曼山菜问题的签名吹式解决方案。
We consider the semi-Riemannian Yamabe type equations of the form \[ -\square u + λu = μ\vert u\vert^{p-1}u\quad\text{ on }M \] where $M$ is either the semi-Euclidean space or the pseudosphere of dimension $m\geq 3$, $\square$ is the semi-Riemannian Laplacian in $M$, $λ\geq0$, $μ\in\mathbb{R}\smallsetminus\{0\}$ and $p>1$. Using semi-Riemannian isoparametric functions on $M$, we reduce the PDE into a generalized Emden-Fowler ODE of the form \[ w''+q(r)w'+λw = μ\vert w\vert^{p-1}w\quad\text{ on } I, \] where $I\subset\mathbb{R}$ is $[0,\infty)$ or $[0,π]$, $q(r)$ blows-up at $0$ and $w$ is subject to the natural initial conditions $w'(0)=0$ in the first case and $w'(0)=w'(π)=0$ in the second. We prove the existence of blowing-up and globally defined solutions to this problem, both positive and sign-changing, inducing solutions to the semi-Riemannian Yamabe type problem with the same qualitative properties, with level and critical sets described in terms of semi-Riemannian isoparametric hypersurfaces and focal varieties. In particular, we prove the existence of sign-changing blowing-up solutions to the semi-Riemannian Yamabe problem in the pseudosphere having a prescribed number of nodal domains.
