Abstract
We consider the following singular semilinear problem
{-Delta u(x) = a(x)u(sigma)(x), x is an element of Omega\{0} (in the distributional sense),
u > 0, on Omega\{0},
lim(vertical bar x vertical bar -> 0) u(x)/ln vertical bar x vertical bar = 0,
u(x) = 0, x is an element of partial derivative Omega,
where sigma < 1; Omega is a bounded regular domain in R-2 with 0 is an element of Omega The weight function a(x) is required to be positive and continuous in Omega\{0} with the possibility to be singular at x = 0 and/or at the boundary partial derivative Omega. 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 <(Omega)over bar>\{0} which could blow-up at 0.