Nonradial singular metrics of constant Q-curvature

In this paper, we study the existence of solutions to the problem(−Δ)n/2u=enuonRn﹨{0},Λ=∫Rnenudx<∞ for n≥3, which corresponds to finding singular conformal metrics gu:=e2u|dx|2 on Rn﹨{0} with constant Q-curvature equal to one and finite volume Λ. We prove that under suitable conditions, the above...

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Bibliographic Details
Published inJournal of mathematical analysis and applications Vol. 526; no. 1; p. 127217
Main Author Wang, Yamin
Format Journal Article
LanguageEnglish
Published Elsevier Inc 01.10.2023
Subjects
Liouville equation
Paneitz operator
Q-curvature
Variational methods
Paneitz operator
Variational methods
Q-curvature
Liouville equation
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Summary:In this paper, we study the existence of solutions to the problem(−Δ)n/2u=enuonRn﹨{0},Λ=∫Rnenudx<∞ for n≥3, which corresponds to finding singular conformal metrics gu:=e2u|dx|2 on Rn﹨{0} with constant Q-curvature equal to one and finite volume Λ. We prove that under suitable conditions, the above equation admits a (nonradial) solution with prescribed polynomial behaviors at zero and infinity. The mass Λ and asymptotic behaviors of solution u can be simultaneously prescribed. This extends results of König-Laurain and Hyder-Mancini-Martinazzi.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2023.127217