论文标题
具有单数集的高阶方程的高阶积极解决方案的局部行为
Local behavior of positive solutions of higher order conformally invariant equations with a singular set
论文作者
论文摘要
我们研究了一些积极解决方案的某些特性，即具有单数集合$$（-Δ）^m u = u^= u^{\ frac {n+2m} {n-2m}} ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ $λ$是$ \ mathbb {r}^n $，$ 1 \ leq m <n/2 $，$ m $的封闭子集和$ m $是整数。当$λ\ subsetω$是一个紧凑型设置时，我们首先为近距离$λ$附近的正面解决方案建立一个渐近爆炸速率估算，其上都有minkowski dimension $ \ overline {\ textmd {dim}}} _ m（λ）_（λ）<\ frac {n-2m} $ k \ leq \ frac {n-2m} {2} $。我们还显示了单数正溶液的渐近对称性假设$λ\ subsetω$是一种平滑的$ k $ dimensional闭合歧管，带有$ k \ leq \ frac \ frac {n-2m} {2} $。最后，当$ω$是整个空间时，获得解决方案的全局对称结果，而$λ$是$ k $ dimensional超平面，带有$ k \ leq \ frac \ frac {n-2m} {2} $。
We study some properties of positive solutions to the higher order conformally invariant equation with a singular set $$ (-Δ)^m u = u^{\frac{n+2m}{n-2m}} ~~~~~~ \textmd{in} ~ Ω\backslash Λ, $$ where $Ω\subset \mathbb{R}^n$ is an open domain, $Λ$ is a closed subset of $\mathbb{R}^n$, $1 \leq m < n/2$ and $m$ is an integer. We first establish an asymptotic blow up rate estimate for positive solutions near the singular set $Λ$ when $Λ\subset Ω$ is a compact set with the upper Minkowski dimension $\overline{\textmd{dim}}_M(Λ) < \frac{n-2m}{2}$, or is a smooth $k$-dimensional closed manifold with $k\leq \frac{n-2m}{2}$. We also show the asymptotic symmetry of singular positive solutions suppose $Λ\subset Ω$ is a smooth $k$-dimensional closed manifold with $k\leq \frac{n-2m}{2}$. Finally, a global symmetry result for solutions is obtained when $Ω$ is the whole space and $Λ$ is a $k$-dimensional hyperplane with $k\leq \frac{n-2m}{2}$.
