Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume

Ali Hyder; Juncheng Wei

doi:10.3934/cpaa.2019123

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2019, Volume 18, Issue 5: 2757-2764. Doi: 10.3934/cpaa.2019123

This issue Previous Article Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc Next Article Existence, uniqueness and regularity of the solution of the time-fractional Fokker–Planck equation with general forcing

Higher order conformally invariant equations in $ {\mathbb R}^3 $ with prescribed volume

Ali Hyder and
Juncheng Wei

Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T1Z2, Canada

Received: October 2018

Revised: January 2019

Published: April 2019

The first author is supported by SNSF Grant No. P2BSP2-172064. The second author is partially supported by NSERC.

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Abstract

In this paper we study the following conformally invariant poly-harmonic equation

$ \Delta^mu = -u^\frac{3+2m}{3-2m}\quad\text{in }\mathbb{R}^3,\quad u>0, $

with $ m = 2,3 $. We prove the existence of positive smooth radial solutions with prescribed volume $ \int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx $. We show that the set of all possible values of the volume is a bounded interval $ (0,\Lambda^*] $ for $ m = 2 $, and it is $ (0,\infty) $ for $ m = 3 $. This is in sharp contrast to $ m = 1 $ case in which the volume $ \int_{\mathbb{R}^3} u^\frac{6}{3-2m}dx $ is a fixed value.

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Mathematics Subject Classification: Primary: 35J30, 35J91, 53A30.

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