Hyder, Ali.
(2016)
* Conformally Euclidean metrics on $\mathbb{R}^n$ with arbitrary total Q-curvature.*
Preprints Fachbereich Mathematik, 2016 (16).

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## Abstract

We study the existence of solution to the problem

\[

(-\Delta)^{n/2} u = Q e^{nu} in \mathbb{R}^n, \kappa := \int_{\mathbb{R}^n} Q e^{nu} dx < \infty,

\]

where $Q \ge 0, \kappa \in (0,\infty)$ and $n \ge 3$. Using ODE techniques Martinazzi for n = 6 and Huang-Ye for n = 4m+2 proved the existence of solution to the above problem with Q = const > 0 for every $\kappa \in (0,\infty)$. We extend these results in every dimension $n \ge 5$, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which Q is non-constant, and under some decay assumptions on Q we can also treat the cases n = 3 and 4.

\[

(-\Delta)^{n/2} u = Q e^{nu} in \mathbb{R}^n, \kappa := \int_{\mathbb{R}^n} Q e^{nu} dx < \infty,

\]

where $Q \ge 0, \kappa \in (0,\infty)$ and $n \ge 3$. Using ODE techniques Martinazzi for n = 6 and Huang-Ye for n = 4m+2 proved the existence of solution to the above problem with Q = const > 0 for every $\kappa \in (0,\infty)$. We extend these results in every dimension $n \ge 5$, thus completely answering the problem opened by Martinazzi. Our approach also extends to the case in which Q is non-constant, and under some decay assumptions on Q we can also treat the cases n = 3 and 4.

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Ehemalige Einheiten Mathematik & Informatik > Analysis (Martinazzi) 12 Special Collections > Preprints Fachbereich Mathematik |
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UniBasel Contributors: | Hyder, Ali |

Item Type: | Preprint |

Publisher: | Universität Basel |

Language: | English |

Last Modified: | 01 Jul 2019 07:48 |

Deposited On: | 28 Mar 2019 09:51 |

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