Liouville theorems for the polyharmonic Hénon-Lane-Emden system

We study Liouville theorems for the following polyharmonic Hénon-Lane-Emden system$$\begin{cases}(-\Delta)^m u = |x|^a v^p \mathrm{in} \mathbb{R}^n, \\(-\Delta)^m v = |x|^b u^q \mathrm{in} \mathbb{R}^n\end{cases}$$when $m, p, q \geq 1, pq \neq 1, a, b \geq 0$. The main conjecture states that $(u, v) = (0, 0)$ is the unique nonnegative solution of this system whenever $(p, q)$ is under the critical Sobolev hyperbola, i.e. $\frac{n+a}{p+1} + \frac{n+b}{q+1} \gt n-2m$. 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$$(-\Delta)^m u = |x|^a u^p \mathrm{in} \mathbb{R}^n$$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.

Keywords

Henon-Lane-Emden system, Liouville theorems, entire solutions, polyharmonic semilinear elliptic equations

2010 Mathematics Subject Classification

35A01, 35A23, 35B08, 35B53, 35J61

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Published 13 August 2014