# Higher Order Conformally Invariant Equations in R^3 with Prescribed Volume

## TL;DR

This paper investigates higher order conformally invariant equations in three-dimensional space, establishing existence of solutions with prescribed volume and characterizing the range of possible volumes for specific cases.

## Contribution

It proves the existence of positive smooth radial solutions with prescribed volume for certain higher order conformally invariant equations in R^3, revealing new volume range properties.

## Key findings

- For m=2, volume range is (0,Λ*], a bounded interval.
- For m=3, volume range is (0,∞).
- Contrasts with m=1 case where volume is fixed.

## 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.