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

In this paper we study the following conformally invariant poly-harmonic equation $${\mathrm{\Delta }}^{m}u=-u^{\frac{3+2m}{3-2m}}\text{in }{\mathbb{R}}^3,u>0,$$ 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 $(0,{\mathrm{\Lambda }}^{\ast }]$ for $m=2$, and it is $(0,\mathrm{\infty })$ 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.