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<p>The existence of weighted nontangential limits and growth rates near a polar set <i>E</i> of positive superharmonic functions satisfying a nonlinear inequality <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(-\Delta u(x)\le cd(x,E)^{-\beta } u(x)^p\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>-</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>c</mi> <mi>d</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>E</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo>-</mo> <mi>β</mi> </mrow> </msup> <mi>u</mi> <msup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mi>p</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and having singularities on <i>E</i> 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.</p>

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

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  • 发表日期：

    2025年12月1日

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  • Kentaro Hirata

摘要

The existence of weighted nontangential limits and growth rates near a polar set E of positive superharmonic functions satisfying a nonlinear inequality \(-\Delta u(x)\le cd(x,E)^{-\beta } u(x)^p\) - 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.