Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations

In this paper, we study singular positive solutions to the semilinear elliptic equation $$-\mathrm{\Delta }u=f(u)\text{ in }\mathrm{\Omega }\setminus \mathrm{\Gamma },$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a bounded or unbounded domain, and $\mathrm{\Gamma }\subset \mathrm{\Omega }$ is a singular closed set with zero Newtonian capacity. When $\mathrm{\Omega }={\mathbb{R}}^n$ and $\mathrm{\Gamma }\subset \{x_1=0\}$, 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 $\mathrm{\Omega }\subset {\mathbb{R}}^n$ and $\mathrm{\Gamma }$ is a smooth closed manifold of dimension $\le n-2$, we show the asymptotic symmetry of singular solutions with respect to the normal direction of $\mathrm{\Gamma }$. 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.