Many solutions for elliptic equations with critical exponents

Let Ω be an open bounded domain in ℝN(N ≥ 3) and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$2^* = \frac{{2N}}{{N - 2}}$$ \end{document}. We are concerned with two kinds of critical elliptic problems. The first one is *\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ - \Delta u - \mu \frac{u}{{\left| x \right|^2 }} = \lambda u + \left| u \right|^{m - 2} u + \theta \left| u \right|^{2^* - 2} u u \in H_0^1 (\Omega ),$$ \end{document} where 0 ∈ Ω, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$0 < \mu < (\frac{{N - 2}}{2})^2 $$ \end{document}, 2 < m < 2* and λ > 0. By using the fountain theorem and concentration estimates, if N ≥ 7 and θ > 0, we establish the existence of infinitely many solutions for the following regularization of (*) with small number ϵ > 0 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ - \Delta u - \mu \frac{u}{{\left| x \right|^2 + \varepsilon }} = \lambda u + \left| u \right|^{m - 2} u + \theta \left| u \right|^{2^* - 2} u u \in H_0^1 (\Omega ).$$ \end{document} Then if θ > 0 is suitably small, we obtain many solutions for problem (*) by taking the process of approximation.