Existence of positive weak solutions with a prescribed singular set of semilinear elliptic equations

In this paper, we consider the problem of the existence of non-negative weak solution u of\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\left\{ \begin{gathered} \Delta u + u^p = 0 in \Omega \hfill \\ u = 0 on \partial \Omega \hfill \\ \end{gathered} \right.$$ \end{document} having a given closed set S as its singular set. We prove that when\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\frac{n}{{n - 2}}< p< \frac{{n + 2\sqrt {n - 1} }}{{n - 4 + 2\sqrt {n - 1} }}$$ \end{document} and S is a closed subset of Ω, then there are infinite many positive weak solutions with S as their singular set. Applying this method to the conformal scalar curvature equation for n ≥ 9, we construct a weak solution\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u \in L^{\frac{{n + 2}}{{n - 2}}} \left( {S^n } \right)$$ \end{document} of\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$L_0 u + L^{\frac{{n + 2}}{{n - 2}}} = 0$$ \end{document} such that Sn is the singular set of u where L0 is the conformal Laplacian with respect to the standard metric of Sn. When n = 4 or 6, this kind of solution has been constructed by Pacard.