Large energy entire solutions for the Yamabe equation. (English) Zbl 1233.35008
The paper under review deals with the construction of finite energy solutions to the Yamabe equation in the whole space \({\mathbb R}^n\). The authors develop an approach which provides examples of non-radial solutions in all dimensions \(n \geq 3\), at the same time providing fine knowledge on the core asymptotic behavior. The main result establishes that for all \(n\geq 3\), there is an entire solution that looks like the soliton \(U\) crowned with \(k\) negative spikes arranged on a regular polygon with radius 1. The proof of this result consists of linearizing the equation around a first approximation and devising an invertibility theory for the linearized operator which takes advantage of the symmetry of the configuration.
Reviewer: Vicenţiu D. Rădulescu (Craiova)
MSC:
| 35B08 | Entire solutions to PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B44 | Blow-up in context of PDEs |
| 35J60 | Nonlinear elliptic equations |
| 35A20 | Analyticity in context of PDEs |
| 35B25 | Singular perturbations in context of PDEs |
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