On the H\'enon-Lane-Emden conjecture

  title={On the H\'enon-Lane-Emden conjecture},
  author={Mostafa Fazly and Nassif Ghoussoub},
  journal={arXiv: Analysis of PDEs},
We consider Liouville-type theorems for the following Henon-Lane-Emden system \hfill -\Delta u&=& |x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. 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… 
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has been studied by a number of authors; see for example [2, 4, 5, 9, 10, 13, 14, 16] and the references therein. In particular in Gidas-Spruck ([5]) it is proved among other things that if f(x, u) =