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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.13622 |
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Table of Contents:
- 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 $(\mathbb{S} ^{2n+1},θ_0)$. 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 $θ_0$ are $C^{0}$-dense among bounded functions which are positive somewhere. Existence results of infinitely many positive solutions to the related equation $-Δ_{\mathbb{H}} u=R(ξ) u^{(n+2) /n}$ 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.