New existence for the higher order prescribed curvature problem

We consider the following prescribed curvature problem involving polyharmonic operator on $ \mathbb{S}^{N}: $

$ \begin{align*} D^{m}u = K(|y|)u^{m^{*}-1}, \quad u > 0 \quad \hbox{in} \quad \mathbb{S}^{N}, \quad u \in H^{m}(\mathbb{S}^{N}), \end{align*} $

where $ K(|y|) $ is a positive function, $ m^{*} = \frac{2N}{N-2m} $ is the critical exponent of Sobolev embedding, $ N>6m. $ $ D^m $ is the $ 2m $ order differential operator given by

$ D^m = \prod\limits_{l = 1}^m (-\Delta_g+\frac{1}{4}(N-2l)(N+2l-2)), $

where $ \Delta_g $ is the Laplace-Beltrami operator on $ \mathbb{S}^N, $ and $ \mathbb{S}^N $ is the unit sphere with Riemann metric $ g. $ By applying the non-degenerate result of positive multi-bubbling solutions constructed in [15], we construct new type solutions by using Lyaponov Schmidt reduction arguments combining with the gluing method. One will see that this type of new solution is difficult to be obtained through direct critical theory and reduction argument. We consider the problem from a different point of view and glue $ n $ bubble solution to $ k $ bubble solution and obtain new type multi bubble solution.