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

Abstract The existence of weighted nontangential limits and growth rates near a polar set E of positive superharmonic functions satisfying a nonlinear inequality $$-\Delta u(x)\le cd(x,E)^{-\beta } u(x)^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>≤</mml:mo> <mml:mi>c</mml:mi> <mml:mi>d</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>p</mml:mi> </mml:msup> </mml:mrow> </mml:math> 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.