Liouville theorems for the polyharmonic Henon-Lane-Emden system

TL;DR

This paper proves Liouville theorems for a class of polyharmonic Hénon-Lane-Emden systems, showing that under certain conditions, the only nonnegative solutions are trivial, especially in odd dimensions and for radial solutions.

Contribution

The paper establishes the uniqueness of nonnegative solutions for the polyharmonic Hénon-Lane-Emden system under subcritical conditions, extending Liouville theorems to higher-order and weighted cases in specific dimensions.

Findings

Liouville theorems hold in dimension n=2m+1 for bounded solutions.

Uniqueness of solutions for radial cases in any dimension.

Nonradial solutions pose significant challenges due to weight functions.

Abstract

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 $m,p,q \ge 1,$ $pq\neq1$, $a,b\ge0$. 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 $|x|^a$ and $|x|^b$ make the problem dramatically more challenging when dealing with nonradial solutions.