Liouville theorems for the polyharmonic Henon-Lane-Emden system

We study Liouville theorems for the following polyharmonic H'{e}non-Lane-Emden system \begin{eqnarray*} \left{\begin{array}{lcl} (-\Delta)^m u&=& |x|^{a}v^p \ \ $\text{in}$\
$\mathbb{R}$^n,\ (-\Delta)^m v&=& |x|^{b}u^q \ \ $\text{in}$\ \ $\mathbb{R}$^n, \end{array}\right. \end{eqnarray*} when $m,p,q\ge 1,$ $pq\mathrm{\ne }1$, $a,b\ge 0$. The main conjecture states that $(u,v)=(0,0)$ is the unique nonnegative solution of this system whenever $(p,q)$ is {\it under} the critical Sobolev hyperbola, i.e. $\frac{n+a}{p+1}+\frac{n+b}{q+1}>n-2m$. We show that this is indeed the case in dimension $n=2m+1$ for bounded solutions. In particular, when $a=b$ and $p=q$, this means that $u=0$ is the only nonnegative bounded solution of the polyharmonic H'{e}non equation \begin{equation*} (-\Delta)^m u= |x|^{a}u^p \ \ $\text{in}$\ \ $\mathbb{R}$^{n} \end{equation*} in dimension $n=2m+1$ provided $p$ is the subcritical Sobolev exponent, i.e., $1<p<1+4m+2a$. Moreover, we show that the conjecture holds for radial solutions in any dimensions. It seems the power weight functions $\mathrm{\mid }x{\mathrm{\mid }}^a$ and $\mathrm{\mid }x{\mathrm{\mid }}^b$ make the problem dramatically more challenging when dealing with nonradial solutions.