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Higher Order Conformally Invariant Equations in R^3 with Prescribed Volume by Hyder, Ali; Wei, Juncheng is a scholarly article available to read on EtoBox.

What is Higher Order Conformally Invariant Equations in R^3 with Prescribed Volume about?

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.

Author

Hyder, Ali; Wei, Juncheng

Published

2019

Language

EN