Solutions to nonlinear Neumann problems with an inverse square potential

Let Ω be an open bounded domain in \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb{R}^N (N\geq3)$$\end{document} with smooth boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial\Omega, 0\in\partial\Omega$$\end{document}. We are concerned with the critical Neumann problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left\{ \begin{array}{ll} -\Delta{u}-\mu\frac{u}{|x|^2}+\lambda u=Q(x)|u|^{2^*-2}{u} \,\,& \quad \mbox{in}\,\,\Omega,\\ \frac{\partial u}{\partial \nu}=0\,\,&\quad \mbox{on}\,\,\partial\Omega, \end{array} \right. (*) $$\end{document}where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 < \mu < \bar{\mu}=(\frac{N-2}{2})^2,\,\,2^*=\frac{2N}{N-2},\,\,\,\,\lambda > 0$$\end{document} and Q(x) is a positive continuous function on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\Omega}$$\end{document}. Using Moser iteration, we give an asymptotic characterization of solutions for (*) at the origin. Under some conditions on Q,  μ, we, by means of a variational method, prove that there exists \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda_0=\lambda_0(\mu) > 0$$\end{document} such that for every \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda > \lambda_0$$\end{document}, problem (*) has a positive solution and a pair of sign-changing solutions.