On the HénonLaneEmden conjecture
Abstract
We consider Liouvilletype theorems for the following HénonLaneEmden 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 nontrivial nonnegative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \frac{N+a}{p+1}+\frac{N+b}{q+1}>{N2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by PhanSouplet in the scalar case. Assuming stability of the solutions, we could then prove Liouvilletype theorems in higher dimensions. For the scalar cases, albeit of second order ($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}{N2}$ (resp., $ 1<p<\frac{N+4+2a}{N4}$). Finally, we show that nonnegative stable solutions of the full HénonLaneEmden system are trivial provided \label{sysdim00} N<2+2(\frac{p(b+2)+a+2}{pq1}) (\sqrt{\frac{pq(q+1)}{p+1}}+ \sqrt{\frac{pq(q+1)}{p+1}\sqrt\frac{pq(q+1)}{p+1}}).
 Publication:

arXiv eprints
 Pub Date:
 July 2011
 arXiv:
 arXiv:1107.5611
 Bibcode:
 2011arXiv1107.5611F
 Keywords:

 Mathematics  Analysis of PDEs
 EPrint:
 Theorem 4 has been added in the new version. 23 pages, Comments are welcome. Updated version  if any  can be downloaded at http://www.birs.ca/~nassif/ or http://www.math.ubc.ca/~fazly/research.html