Singular solutions of a nonlinear equation in a punctured domain of <inline-formula><tex-math id="M1">\begin{document}$\mathbb{R}^{2}$\end{document}</tex-math></inline-formula>

Abstract

We consider the following singular semilinear problemwhere σ < 1, Ω is a bounded regular domain in R 2 with 0 ∈ Ω. The weight function a(x) is required to be positive and continuous in Ω\{0} with the possibility to be singular at x = 0 and/or at the boundary ∂Ω. When the function a satisfies sharp estimates related to Karamata class, we prove the existence and global asymptotic behavior of a positive continuous solution on Ω\{0} which could blow-up at 0.