AI 中文总结
本文研究半线性椭圆方程奇异正解，在奇异集为零容量或低维流形时，建立了解的对称性、单调性和渐近对称性，并改进了已有结果，采用改进的移动球面方法简化证明。
AI 中文摘要
本文研究半线性椭圆方程$$ - \Delta u = f(u) ~~~~~~ \textmd{in} ~ \Omega \setminus \Gamma, $$的奇异正解，其中$\Omega \subset \R^n$是有界或无界区域，$\Gamma \subset \Omega$是牛顿容量为零的奇异闭集。当$\Omega = \R^n$且$\Gamma \subset \{ x_1 = 0 \}$时，我们建立了奇异解关于超平面$\{ x_1 = 0 \}$的对称性以及它们在$x_1$方向上的单调性。对于$\Omega \subset \R^n$且$\Gamma$是维数$\leq n - 2$的光滑闭流形的情形，我们证明了奇异解关于$\Gamma$的法线方向的渐近对称性。这些结果显著改进了Chen-Lin（Duke Math. J. 1995；Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2001）、Li（Invent. Math. 1996）和Sciunzi（J. Math. Pures Appl. 2017）的结果。与他们的证明不同，我们采用了改进版的移动球面方法，这也简化了分析并扩展了其适用性。
英文摘要
In this paper, we study singular positive solutions to the semilinear elliptic equation $$ - Δu = f(u) ~~~~~~ \textmd{in} ~ Ω\setminus Γ, $$ where $Ω\subset \R^n$ is a bounded or unbounded domain, and $Γ\subset Ω$ is a singular closed set with zero Newtonian capacity. When $Ω= \R^n$ and $Γ\subset \{ x_1 = 0 \}$, we establish the symmetry of singular solutions with respect to the hyperplane $\{ x_1 = 0 \}$ and their monotonicity in the $x_1$-direction. For the case where $Ω\subset \R^n$ and $Γ$ is a smooth closed manifold of dimension $\leq n - 2$, we show the asymptotic symmetry of singular solutions with respect to the normal direction of $Γ$. These results significantly improve those of Chen-Lin (Duke Math. J. 1995; Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2001), Li (Invent. Math. 1996) and Sciunzi (J. Math. Pures Appl. 2017). Unlike their proofs, we employ an improved version of the moving sphere method, which also simplifies the analysis and extends its applicability.
Comments34 pages. Fixed some typos
