A remark on singular solutions of semilinear elliptic equations in two dimensions
Abstract
We study positive radial singular solutions of −Δu=f(u) in two space dimensions. Fujishima, Ioku, Ruf, and Terraneo constructed such solutions under asymptotic assumptions on f and an additional convergence-rate condition, denoted by (f3). We prove their local existence result, with the same approximate solution and remainder estimate, under the original assumptions (f1) and (f2) alone. The proof combines a contraction on finite intervals with a nonlinear energy estimate uniform in the terminal point and requires only C2 regularity of f near infinity. We also obtain derivative asymptotics and, under additional global assumptions, a singular Dirichlet solution which solves the equation distributionally across the origin. An analytic oscillatory example satisfies our hypotheses but violates both (f3) and the derivative assumptions of Kikuchi and Kumagai. For every fixed finite iterate of the exponential of order at least two, we verify the hypotheses and quantify the remainder; condition (f3) fails for all iterates of order at least three.
Related
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BibTeX
@misc{fujishima2026remark,
title = {{A remark on singular solutions of semilinear elliptic equations in two dimensions}},
author = {Y. Fujishima and N. Ioku and B. Ruf and E. Terraneo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04520},
eprint = {2611.04520},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04520},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - A remark on singular solutions of semilinear elliptic equations in two dimensions
AU - Y. Fujishima
AU - N. Ioku
AU - B. Ruf
AU - E. Terraneo
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04520
UR - https://arxiv.org/abs/2611.04520
AB - We study positive radial singular solutions of −Δu=f(u) in two space dimensions. Fujishima, Ioku, Ruf, and Terraneo constructed such solutions under asymptotic assumptions on f and an additional convergence-rate condition, denoted by (f_3). We prove their local existence result, with the same approximate solution and remainder estimate, under the original assumptions (f_1) and (f_2) alone. The proof combines a contraction on finite intervals with a nonlinear energy estimate uniform in the terminal point and requires only C^2 regularity of f near infinity. We also obtain derivative asymptotics and, under additional global assumptions, a singular Dirichlet solution which solves the equation distributionally across the origin. An analytic oscillatory example satisfies our hypotheses but violates both (f_3) and the derivative assumptions of Kikuchi and Kumagai. For every fixed finite iterate of the exponential of order at least two, we verify the hypotheses and quantify the remainder; condition (f_3) fails for all iterates of order at least three.
ER -