Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent (Q1902013)

The authors study the semilinear equation \(-\Delta u= u^{(n+ 2)/(n- 2)}\) in \(\Omega\backslash Z\), where \(\Omega\) is an open domain in \(\mathbb{R}^n\) and \(Z\) a closed set. Their aim is to extend the results of Caffarelli, Gidas and Spruck who studied the equation on \(\mathbb{R}^n\backslash \{0\}\). Other nonlinearities \(f(u)\), that are close to this critical \(u^{(n+ 2)/(n- 2)}\), are also considered. The first result they prove is the following. If \(\text{Cap}(Z)= 0\) then positive solutions satisfy \(u(x)\leq Cd(x, Z)^{-(n- 2)/2}\) on \(B_R\) if \(B_{2R}\subset \Omega\). They claim that their approach will provide a simpler way to obtain the asymptotic result of Caffarelli, Gidas and Spruck. A consecutive paper in this direction is announced.