论文标题
临界拉普拉斯方程和RICCI曲率的注释
A note on the critical Laplace Equation and Ricci curvature
论文作者
论文摘要
我们研究了严格的积极解决方案\ [ - Δu= n（n-2）u^{\ frac {n+2} {n+2} {n-2}}，\]，最喜欢$ d（o，x）^{ - （n-2）/2} $，在完全的nonnontonnonmonnonparpact $（m，n nontocation $（n nontocation）$（o，x）^{ - （n-2）/2 \ geq 3 $。我们证明，在对体积增长的额外假设下，不存在这种解决方案，除非$（m，g）$是$ \ mathbb {r}^n $和$ u $的等级，而$ u $是Talenti功能。该方法对沿$ u $的级别集进行了适当定义的合适函数进行基本分析。
We study strictly positive solutions to the critical Laplace equation \[ - Δu = n(n-2) u^{\frac{n+2}{n-2}}, \] decaying at most like $d(o, x)^{-(n-2)/2}$, on complete noncompact manifolds $(M, g)$ with nonnegative Ricci curvature, of dimension $n \geq 3$. We prove that, under an additional mild assumption on the volume growth, such a solution does not exist, unless $(M, g)$ is isometric to $\mathbb{R}^n$ and $u$ is a Talenti function. The method employs an elementary analysis of a suitable function defined along the level sets of $u$.
