A remark on singular solutions of semilinear elliptic equations in two dimensions

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.