The Non-degenerate Condition for Prescribed Q Curvature Problem on \(\mathbb {S}^n\)

For \(n\ge 2\) $n\ge 2$ , we consider the equation 0.1 \(\begin{aligned} P_{\mathbb {S}^n}u+2Q_{g_{\mathbb {S}^n}}=2Q_ge^{nu}\qquad \textrm{on}\qquad \mathbb {S}^n. \end{aligned}\) $$\begin{array}{c}\hfill P_{{\mathbb{S}}^n}u+2Q_{g_{{\mathbb{S}}^n}}=2Q_g{e}^{\mathit{nu}}\text{on}{\mathbb{S}}^n.\end{array}$$ In general, we always assume the non-degenerate condition for Eq. (0.1), i.e., 0.2 \(\begin{aligned} |\nabla _{g_{\mathbb {S}^n}} Q_g|^2+\left( \Delta _{g_{\mathbb {S}^n}} Q_g\right) ^2\ne 0\quad \textrm{on}\quad \mathbb {S}^n. \end{aligned}\) $$\begin{array}{c}\hfill |{\mathrm{\nabla }}_{g_{{\mathbb{S}}^n}}{Q}_g{|}^2+{\left({\mathrm{\Delta }}_{g_{{\mathbb{S}}^n}},Q_g\right)}^2\ne 0\text{on}{\mathbb{S}}^n.\end{array}$$ 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.