Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations
Abstract
In this paper, we study singular positive solutions to the semilinear elliptic equation − Δu = f(u) textmdin Ω∖ Γ, where Ω⊂ Rn is a bounded or unbounded domain, and Γ⊂ Ω is a singular closed set with zero Newtonian capacity. When Ω= Rn and Γ⊂ { x1 = 0 }, we establish the symmetry of singular solutions with respect to the hyperplane { x1 = 0 } and their monotonicity in the x1-direction. For the case where Ω⊂ Rn and Γ is a smooth closed manifold of dimension ≤ n − 2, we show the asymptotic symmetry of singular solutions with respect to the normal direction of Γ. These results significantly improve those of Chen-Lin (Duke Math. J. 1995; Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2001), Li (Invent. Math. 1996) and Sciunzi (J. Math. Pures Appl. 2017). Unlike their proofs, we employ an improved version of the moving sphere method, which also simplifies the analysis and extends its applicability.
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BibTeX
@misc{du2026symmetry,
title = {{Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations}},
author = {Xusheng Du and Hui Yang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16516},
eprint = {2609.16516},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16516},
}
RIS
TY - UNPB
TI - Symmetry, monotonicity, and asymptotics of singular solutions to semilinear elliptic equations
AU - Xusheng Du
AU - Hui Yang
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.16516
UR - https://arxiv.org/abs/2609.16516
AB - In this paper, we study singular positive solutions to the semilinear elliptic equation − Δu = f(u) textmdin Ω∖ Γ, where Ω⊂ R^n is a bounded or unbounded domain, and Γ⊂ Ω is a singular closed set with zero Newtonian capacity. When Ω= R^n and Γ⊂ { x_1 = 0 }, we establish the symmetry of singular solutions with respect to the hyperplane { x_1 = 0 } and their monotonicity in the x_1-direction. For the case where Ω⊂ R^n and Γ is a smooth closed manifold of dimension ≤ n − 2, we show the asymptotic symmetry of singular solutions with respect to the normal direction of Γ. These results significantly improve those of Chen-Lin (Duke Math. J. 1995; Ann. Scuola Norm. Sup. Pisa Cl. Sci. 2001), Li (Invent. Math. 1996) and Sciunzi (J. Math. Pures Appl. 2017). Unlike their proofs, we employ an improved version of the moving sphere method, which also simplifies the analysis and extends its applicability.
ER -