Weighted nontangential limits on a polar set of superharmonic functions satisfying a nonlinear inequality

The existence of weighted nontangential limits and growth rates near a polar set E of positive superharmonic functions satisfying a nonlinear inequality -Δ u(x)≤ cd(x,E)-β u(x)ᵖ - Δ u ( x ) ≤ c d ( x , E ) - β u ( x ) p and having singularities on E are investigated. A main result extends Lions’ result (1980) regarding the asymptotic behavior near an isolated singularity of a positive solution of the Lane–Emden equation to the case of non-isolated singularities, and complements the author and Ono’s result (2014) regarding removable singularities of solutions of semilinear elliptic equations.