Two Disjoint and Infinite Sets of Solutions for an Elliptic Equation Involving Critical Hardy-Sobolev Exponents

In this paper, by an approximating argument, we obtain two disjoint and infinite sets of solutions for the following elliptic equation with critical Hardy-Sobolev exponents \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ {\matrix{{ - \Delta u = \mu |u{|^{{2^ * } - 2}}u + {{|u{|^{{2^ * }(s) - 2}}u} \over {|x{|^s}}} + a(x)|u{|^{q - 2}}u} \hfill & {{\rm{in}}\,\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm{on}}\,\,\partial \Omega,} \hfill \cr } } \right.$$\end{document} where Ω is a smooth bounded domain in ℝN with 0 ∈ ∂Ω and all the principle curvatures of ∂Ω at 0 are negative, a∈C1(Ω¯,ℝ∗+)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in {{\cal C}^1}(\bar \Omega,{\mathbb{R}^{ * + }})$$\end{document}, μ > 0, 0 < s < 2, 1< q < 2 and N>2q+1q−1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N > 2{{q + 1} \over {q - 1}}$$\end{document}. By 2∗:=2NN−2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^ * }: = {{2N} \over {N - 2}}$$\end{document} and 2∗(s):=2(N−s)N−2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${2^ * }(s): = {{2(N - s)} \over {N - 2}}$$\end{document} we denote the critical Sobolev exponent and Hardy-Sobolev exponent, respectively.