On the CR Nirenberg problem: density and multiplicity of solutions

TL;DR

This paper investigates the density and multiplicity of positive solutions to the prescribed Webster scalar curvature problem on CR spheres, constructing multi-bump solutions and demonstrating density results using advanced variational and analytical techniques.

Contribution

It introduces new methods to construct arbitrarily many multi-bump solutions and proves the density of certain curvature functions among positive functions on CR spheres.

Findings

Existence of arbitrarily many multi-bump solutions.

Webster scalar curvature functions are dense among positive functions.

Infinitely many solutions exist for related equations on the Heisenberg group.

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 $(\mathbb{S} ^{2n+1},\theta_0)$. 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.