On the CR Nirenberg problem: density and multiplicity of solutions

2024/04/21 by Tang, Zhongwei, Wang, Heming, Zhang, Bingwei
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2404.13622

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 (\mathbbS 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 -Δ_ℍ u=R(ξ) u^{(n+2) /n} on the Heisenberg group \Hn 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.

Related

• On the density and multiplicity of solutions to the fractional Nirenberg problem
• Singular CR structures of constant Webster curvature and applications
• On the Webster scalar curvature problem on the CR sphere with a cylindrical-type symmetry
• On the Nirenberg problem on spheres: Arbitrarily many solutions in a perturbative setting
• The Nirenberg problem and its generalizations: A unified approach