On the CR Nirenberg problem: density and multiplicity of solutions

Abstract

We prove some results on the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on the $(2n+1)$-dimensional standard unit CR sphere $({\mathbb{S}}^{2n+1},\hat{I}{¸}_0)$. Specifically, we construct arbitrarily many multi-bump solutions via the variational gluing method. In particular, we show the Webster scalar curvature functions of contact forms conformal to $\hat{I}{¸}_0$ are $C^0$-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation $-\hat{I}{”}_{\mathbb{H}}u=R(\hat{I}¾)u^{(n+2)/n}$ on the Heisenberg group $\Hn $ with $R(\hat{I}¾)$ being asymptotically periodic with respect to left translation are also obtained. Our proofs make use of a refined analysis of bubbling behavior, gradient flow, Pohozaev identity, as well as blow up arguments.