The aim of this paper is to establish the existence and global asymptotic behavior of a positive continuous solution for the following semilinear problem: \t\t\t{−Δu(x)=a(x)uσ(x),x∈D,u>0,in D,u(x)=0,x∈∂D,lim|x|→∞u(x)ln|x|=0,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta u(x)=a(x)u^{\\sigma }(x), \\quad x\\in D, \\\\ u>0, \\quad \\text{in }D, \\\\ u(x)=0, \\quad x\\in \\partial D, \\\\ \\lim_{ \\vert x \\vert \\rightarrow \\infty }\\frac{u(x)}{ \\ln \\vert x \\vert }=0, \\end{cases} $$\\end{document} where sigma <1, D is an unbounded domain in mathbb{R}^{2} with a compact nonempty boundary ∂D consisting of finitely many Jordan curves. As main tools, we use Kato class, Karamata regular variation theory and the Schauder fixed point theorem.