Liouville theorems for the polyharmonic Henon-Lane-Emden system

Abstract

We study Liouville theorems for the following polyharmonic Hénon-Lane-Emden system \begin{eqnarray*} \left\{\begin{array}{lcl} (-Δ)^m u&=& |x|^{a}v^p \ \ \text{in}\ \ \mathbb{R}^n,\\ (-Δ)^m v&=& |x|^{b}u^q \ \ \text{in}\ \ \mathbb{R}^n, \end{array}\right. \end{eqnarray*} when $m,p,q\ge 1,$ $pq\ne 1$, $a,b\ge 0$. 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 Hénon equation \begin{equation*} (-Δ)^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 $|x{|}^a$ and $|x{|}^b$ make the problem dramatically more challenging when dealing with nonradial solutions.

arXiv comment16 pages. Submitted. This is an extension of the work of Nassif Ghoussoub and the author entitled "On the Hénon-Lane-Emden conjecture" given in arXiv:1107.5611 to polyharmonic equations and systems