On Dancer-type solutions for the Lane--Emden equation via semivirial-vanishing geometry
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| Format: | Preprint |
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2026
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| _version_ | 1866914621818929152 |
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| author | Luo, Yongming |
| author_facet | Luo, Yongming |
| contents | Aubin--Talenti bubbles describe the decaying positive solutions of the zero-frequency critical Lane--Emden equation in Euclidean space. By appealing to bifurcation methods, Dancer constructed in his seminar paper \cite{DancerSolution} positive-frequency solutions to the Lane--Emden equation which decay in the noncompact directions and are periodic in one direction. Alternatively, we give in this paper an energy-based variational construction of such Dancer-type solutions via the semivirial-vanishing geometry developed in author's recent work for studying focusing NLS on waveguide manifolds. The main new ingredient is a strict sub-bubbling estimate below the Euclidean Sobolev threshold. Unlike the usual Brezis--Nirenberg mechanism, no lower-order focusing perturbation is available in our model. Instead, the energy drop is produced by the bounded periodic direction: truncating a Euclidean bubble to one period removes a leading-order part of the gradient tail, while the nonlinear tail is of lower order. This restores compactness of minimizing sequences and yields normalized ground states for every prescribed mass, thereby answering an open question from \cite{Luo_energy_crit}. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01692 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Dancer-type solutions for the Lane--Emden equation via semivirial-vanishing geometry Luo, Yongming Analysis of PDEs Aubin--Talenti bubbles describe the decaying positive solutions of the zero-frequency critical Lane--Emden equation in Euclidean space. By appealing to bifurcation methods, Dancer constructed in his seminar paper \cite{DancerSolution} positive-frequency solutions to the Lane--Emden equation which decay in the noncompact directions and are periodic in one direction. Alternatively, we give in this paper an energy-based variational construction of such Dancer-type solutions via the semivirial-vanishing geometry developed in author's recent work for studying focusing NLS on waveguide manifolds. The main new ingredient is a strict sub-bubbling estimate below the Euclidean Sobolev threshold. Unlike the usual Brezis--Nirenberg mechanism, no lower-order focusing perturbation is available in our model. Instead, the energy drop is produced by the bounded periodic direction: truncating a Euclidean bubble to one period removes a leading-order part of the gradient tail, while the nonlinear tail is of lower order. This restores compactness of minimizing sequences and yields normalized ground states for every prescribed mass, thereby answering an open question from \cite{Luo_energy_crit}. |
| title | On Dancer-type solutions for the Lane--Emden equation via semivirial-vanishing geometry |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2606.01692 |