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 (S ²ⁿ⁺¹,θ₀). 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 θ₀ are C⁰-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation -ΔH u=R(ξ) u(n+2) /n on the Heisenberg group with R(ξ) 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.