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

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 \{