Elliptic equation with a singular potential in a domain with a conic point

This paper deals with the behavior of the nonnegative solutions of the problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$- \Delta u = V(x)u, \left. u \right|\partial \Omega = \varphi (x)$$\end{document} in a conical domain Ω ⊂ ℝn, n ≥ 3, where 0 ≤ V (x) ∈ L1(Ω), 0 ≤ ϕ(x) ∈ L1(∂Ω) and ϕ(x) is continuous on the boundary ∂Ω. It is proved that there exists a constant C*(n) = (n − 2)2/4 such that if V0(x) = (c + λ1)|x|−2, then, for 0 ≤ c ≤ C*(n) and V(x) ≤ V0(x) in the domain Ω, this problem has a nonnegative solution for any nonnegative boundary function ϕ(x) ∈ L1(∂Ω); for c > C*(n) and V(x) ≥ V0(x) in Ω, this problem has no nonnegative solutions if ϕ(x) > 0.