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Multipeak solutions for fractional elliptic equations on conformal infinities

Part of: Partial differential equations on manifolds; differential operators Partial differential equations Global differential geometry Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  19 December 2025

Yuan Gao Open the ORCID record for Yuan Gao [Opens in a new window] ,
Yuxia Guo  and
Ning Zhou
Yuan Gao*
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China (gaoy22@mails.tsinghua.edu.cn)
Yuxia Guo
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing, P. R. China (yguo@tsinghua.edu.cn)
Ning Zhou
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin, P. R. China (zhouning@nankai.edu.cn)
*
*Corresponding author.
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Abstract

We consider the following problem

\begin{equation*}\varepsilon^{2 s} P_{\hat{h}}^s u+u=u^p, \quad u \gt 0 \quad \text {on } \,(M, \hat{h}),\end{equation*}
on an asymptotically hyperbolic manifold $(X, g^{+})$ with conformal infinity $(M,[\hat{h}])$, where $s\in (0,1)$, $P_{\hat{h}}^s$ is the fractional conformally invariant operators, $1 \lt p \lt \frac{n+2s}{n-2s}$. By Lyapunov–Schmidt reduction method, we prove the existence of solutions whose peaks collapse, as $\varepsilon$ goes to zero, to a $C^1$-stable critical point of the mean curvature $H$ for $0 \lt s \lt {1}/{2}$ or a $C^1$-stable critical point of a function involving the scalar curvature and the second fundamental form for ${1}/{2}\le s \lt 1$.

Keywords

Lyapunov–Schmidt reductionmultipeak solutionsfractional conformally invariant operators

MSC classification

Primary: 35B09: Positive solutions
Secondary: 35B44: Blow-up 35R11: Fractional partial differential equations 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions 58J05: Elliptic equations on manifolds, general theory

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 55
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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