Existence and density results of conformal metrics with prescribed higher order Q-curvature on Sn

Abstract

We prove some results on the density and multiplicity of positive solutions to the conformal Q-curvature equations $P_m\left(v\right)=K{v}^{\frac{n+2m}{n-2m}}$ on the n-dimensional standard unit sphere $(S^n,g_0)$ for all $m\in [1,\frac{n}{2})$ and m is an integer, where $P_m$ is the intertwining operator of order 2m and K is the prescribed Q-curvature function. More specifically, by using the variational gluing method, refined analysis of bubbling behavior, Pohozaev identity, as well as the blow up argument for nonlinear integral equations, we construct arbitrarily many multi-bump solutions. In particular, we show the smooth positive Q-curvature functions of metrics conformal to $g_0$ are dense in the $C^0$ topology. Existence results of infinitely many positive solutions to the poly-harmonic equations ${(-\Delta )}^{m}u=K\left(x\right)u^{\frac{n+2m}{n-2m}}$ in $R^n$ with $K\left(x\right)$ being asymptotically periodic are also obtained.