Non-Degeneracy of \(\mathcal {O}(3)\) Invariant Solutions for Higher Order Prescribed Curvature Problem and Applications

We consider the following prescribed curvature problem involving polyharmonic operators on \(\mathbb {S}^{N}\) ${\mathbb{S}}^N$ \(\begin{aligned} D^{m}{\tilde{u}} = \widetilde{K}(y){\tilde{u}}^{m^{*}-1}, \quad {\tilde{u}} > 0 \ \hbox { in } \mathbb {S}^{N}, \quad {\tilde{u}} \in H^{m}(\mathbb {S}^{N}), \end{aligned}\) $$\begin{array}{c}\hfill D^m\tilde{u}=\tilde{K}(y){\tilde{u}}^{m^{\ast }-1},\tilde{u}>0\text{in}{\mathbb{S}}^N,\tilde{u}\in H^m({\mathbb{S}}^N),\end{array}$$ where \(\widetilde{K}(y)>0\) $\tilde{K}(y)>0$ is a radial function, \(m^{*}=\frac{2N}{N-2m}, m\ge 1\) $m^{\ast }=\frac{2N}{N-2m},m\ge 1$ is an integer and \(D^m\) $D^m$ is 2m-order differential operator given by \(\begin{aligned} D^m=\prod _{i=1}^m \bigg (-\Delta _g+\frac{1}{4}(N-2i)(N+2i-2)\bigg ). \end{aligned}\) $$\begin{array}{c}\hfill D^m=\prod _{i=1}^m(-{\mathrm{\Delta }}_g+\frac{1}{4}(N-2i)(N+2i-2)).\end{array}$$ Here \(\Delta _g\) ${\mathrm{\Delta }}_g$ is the Laplace-Beltrami operator on \(\mathbb {S}^N\) ${\mathbb{S}}^N$ , and \(\mathbb {S}^N\) ${\mathbb{S}}^N$ is the unit sphere with Riemann metric g. We are concerned with the solutions which are invariant under some non-trivial sub-group of \(\mathcal {O}(3)\) $\mathcal{O}(3)$ to the above problem. We first prove a non-degeneracy result for this kind of \(\mathcal {O}(3)\) $\mathcal{O}(3)$ invariant solutions. As an application, we consider an eigenvalue problem, we investigate the properties of the eigenvalues and obtain the Morse index estimate of the \(\mathcal {O}(3)\) $\mathcal{O}(3)$ invariant solutions. Our proof is based on the local Pohozaev identities, blow-up analysis, and the properties of the Green function.