Existence of a Positive Solution for a Class of Elliptic Problems in Exterior Domains Involving Critical Growth

In this work, we use variational methods to prove the existence of a positive solution for the following class of elliptic problems, -Δu+u=uq+ϵu2∗-1,inΩ,u>0,inΩ,u∈H01(Ω),(Pϵ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{array}{ll}-\Delta u + u = u^q + \epsilon u^{2^*-1}, \, \,{\rm in}\, \Omega,\\ u > 0, \,\,{\rm in}\,\, \Omega,\\ u \in H^1_0(\Omega),\end{array}\right. \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (P_\epsilon) $$\end{document} where Ω⊂RN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Omega \subset \mathbb{R}^N}$$\end{document} is an exterior domain, N≥3,1<q<2∗-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${N \geq 3, 1 < q < 2^* - 1}$$\end{document} and ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\epsilon}$$\end{document} is a small positive parameter.