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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2404.13622 |
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| _version_ | 1866913323499388928 |
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| author | Tang, Zhongwei Wang, Heming Zhang, Bingwei |
| author_facet | Tang, Zhongwei Wang, Heming Zhang, Bingwei |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_13622 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the CR Nirenberg problem: density and multiplicity of solutions Tang, Zhongwei Wang, Heming Zhang, Bingwei Analysis of PDEs 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. |
| title | On the CR Nirenberg problem: density and multiplicity of solutions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2404.13622 |