Infinitely many bubbling solutions and non-degeneracy results to fractional prescribed curvature problems

We consider the following fractional prescribed curvature problem $$(-\Delta)^s u= K(y)u^{2^*_s-1},\ \ u>0,\ \ y\in \mathbb{R}^N,\qquad (0.1)$$ where \(s\in(0,\frac{1}{2})\) for \(N=3\), \(s\in(0,1)\) for \(N\geqslant4\) and \(2^*_s=\frac{2N}{N-2s}\) is the fractional critical Sobolev exponent, \...

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Bibliographic Details
Published in:arXiv.org 2022-08
Main Authors: Duan, Lixiu, Guo, Qing
Format: Article
Language:English
Subjects:
Curvature
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Summary:We consider the following fractional prescribed curvature problem $$(-\Delta)^s u= K(y)u^{2^*_s-1},\ \ u>0,\ \ y\in \mathbb{R}^N,\qquad (0.1)$$ where \(s\in(0,\frac{1}{2})\) for \(N=3\), \(s\in(0,1)\) for \(N\geqslant4\) and \(2^*_s=\frac{2N}{N-2s}\) is the fractional critical Sobolev exponent, \(K(y)\) has a local maximum point in \(r\in(r_0-\delta,r_0+\delta)\). First, for any sufficient large \(k\), we construct a \(2k\) bubbling solution to (0.1) of some new type, which concentrate on an upper and lower surfaces of an oblate cylinder through the Lyapunov-Schmidt reduction method. Furthermore, a non-degeneracy result of the multi-bubbling solutions is proved by use of various Pohozaev identities, which is new in the study of the fractional problems.
ISSN:2331-8422