Hyder, Ali.
(2015)
* Existence of entire solutions to a fractional Liouville equation in R^n.*
Preprints Fachbereich Mathematik, 2015 (01).

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## Abstract

We study the existence of solutions to the problem

\[

(−∆)^{n/2} u = Qe^{nu} in \mathbb{R}^n, V := \int_{\mathbb{R}^n} e^{nu} dx < ∞,

\]

where Q = (n − 1)! or Q = −(n − 1)!. Extending the works of Wei-Ye and Hyder-Martinazzi to arbitrary odd dimension n ≥ 3 we show that to a certain extent the asymptotic behavior of u and the constant V can be prescribed simultaneously. Furthermore if Q = −(n − 1)! then V can be chosen to be any positive number. This is in contrast to the case n = 3, Q = 2, where Jin-Maalaoui-Martinazzi-Xiong showed that necessarily V ≤ |S^3|, and to the case n=4, Q=6, where C-S. Lin showed that V ≤|S^4|.

\[

(−∆)^{n/2} u = Qe^{nu} in \mathbb{R}^n, V := \int_{\mathbb{R}^n} e^{nu} dx < ∞,

\]

where Q = (n − 1)! or Q = −(n − 1)!. Extending the works of Wei-Ye and Hyder-Martinazzi to arbitrary odd dimension n ≥ 3 we show that to a certain extent the asymptotic behavior of u and the constant V can be prescribed simultaneously. Furthermore if Q = −(n − 1)! then V can be chosen to be any positive number. This is in contrast to the case n = 3, Q = 2, where Jin-Maalaoui-Martinazzi-Xiong showed that necessarily V ≤ |S^3|, and to the case n=4, Q=6, where C-S. Lin showed that V ≤|S^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: | 09 May 2019 08:56 |

Deposited On: | 28 Mar 2019 09:51 |

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