  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.
