Multiplicity of Solutions and Degenerate Solutions of Yamabe-Type Equations on Projective Spaces

We consider Yamabe-type equations on Projective Spaces $\mathbb{C}{\mathbf{P}}^n$ and $\mathbb{H}{\mathbf{P}}^n$ with the respectives canonical metrics, and study the existence and multiplicity of solutions of Yamabe-type equation, which are invariant by cohomogeneity one actions of $U(n)$ and $Sp(n)$ respectively. We also prove the existence of degenerate solutions of the Yamabe-type equationn on $\mathbb{C}{\mathbf{P}}^n$ and $\mathbb{H}{\mathbf{P}}^n$, which are invariant by cohomogeneity one actions of $U(n)$ and $Sp(n)$ respectively.