On the CR Nirenberg problem: density and multiplicity of solutions

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 $(\mathbb{S} ^{2n+1},\theta_0)$. 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 $\theta_0$ are $C^{0}$-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation $-\Delta_{\mathbb{H}} u=R(\xi) u^{(n+2) /n}$ on the Heisenberg group $\Hn$ with $R(\xi)$ 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.