The Dirichlet problem for semilinear elliptic equations in an infinite sector

We consider the Dirichlet problem for a semilinear elliptic equation in an unbounded sectorial domain Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega $$\end{document} of the two-dimensional space. The problem is supplemented with a limiting behavior related to a prescribed root z of the nonlinearity of the equation. The one-dimensional setting of the problem has a unique solution Vz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{z}$$\end{document}, and the goal of the present paper is to construct a two-dimensional positive and bounded solution u approaching Vz(d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$V_{z}(d)$$\end{document} when d=d(x,∂Ω)→∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d = d(x,\partial \Omega )\rightarrow \infty $$\end{document}. This is established using sub- and supersolutions method and employing a sliding argument.