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

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

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  • 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.

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

    Citation:

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