On the Hénon-Lane-Emden conjecture

Mostafa Fazly; Nassif Ghoussoub

doi:10.3934/dcds.2014.34.2513

Article Contents

2014, Volume 34, Issue 6: 2513-2533. Doi: 10.3934/dcds.2014.34.2513

This issue Previous Article Some symmetry results for entire solutions of an elliptic system arising in phase separation Next Article The role of the scalar curvature in some singularly perturbed coupled elliptic systems on Riemannian manifolds

On the Hénon-Lane-Emden conjecture

Mostafa Fazly^1, and
Nassif Ghoussoub^2,

1.
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
2.
Department of Mathematics, The University of British Columbia, Vancouver BC Canada V6T 1Z2

Received: September 2012

Revised: May 2013

Published: June 2014

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Abstract

We consider Liouville-type theorems for the following Hénon-Lane-Emden system \begin{eqnarray*} \left\{ \begin{array}{lcl} -\Delta u&=& |x|^{a}v^p \ \ in\ \ \mathbb{R}^n,\\ -\Delta v&=& |x|^{b}u^q \ \ in\ \ \mathbb{R}^n, \end{array}\right. \end{eqnarray*} when $p,q \ge 1,$ $pq\neq1$, $a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \frac{n+a}{p+1}+\frac{n+b}{q+1}>{n-2}$. We show that this is indeed the case in dimension $n=3$ provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case.
Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $n\ge 3$ in the first case (resp., $n\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1< p < \frac{n+2+2a}{n-2}$ (resp., $ 1< p < \frac{n+4+2a}{n-4}$). Finally, we show that non-negative stable solutions of the full Hénon-Lane-Emden system are trivial provided \begin{equation*}\label{sysdim00} n < 2 + 2 (\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}} + \sqrt{ \frac{pq(q+1)}{p+1} - \sqrt{\frac{pq(q+1)}{p+1}}}). \end{equation*}

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Mathematics Subject Classification: 35J47, 35B33, 35B45, 35B08.

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