The Non-degenerate Condition for Prescribed Q Curvature Problem on Sn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {S}^n$$\end{document}

For n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n\ge 2$$\end{document}, we consider the equation 0.1PSnu+2QgSn=2QgenuonSn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} P_{\mathbb {S}^n}u+2Q_{g_{\mathbb {S}^n}}=2Q_ge^{nu}\qquad \textrm{on}\qquad \mathbb {S}^n. \end{aligned}$$\end{document}In general, we always assume the non-degenerate condition for Eq. (0.1), i.e., 0.2|∇gSnQg|2+ΔgSnQg2≠0onSn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} |\nabla _{g_{\mathbb {S}^n}} Q_g|^2+\left( \Delta _{g_{\mathbb {S}^n}} Q_g\right) ^2\ne 0\quad \textrm{on}\quad \mathbb {S}^n. \end{aligned}$$\end{document}However, in this paper, we provide some nontrivial examples that do not satisfy condition (0.2); nevertheless, Eq. (0.1) still has at least one solution.