We consider Liouville-type theorems for the following H\'{e}non-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≥0. The main conjecture states that there is no non-trivial
non-negative solution whenever (p,q) is under the critical Sobolev hyperbola,
i.e. p+1N+a+q+1N+b>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≥0=b and p>1=q), we show that for all dimensions N≥3 in the
first case (resp., N≥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<N−2N+2+2a (resp., 1<p<N−4N+4+2a).
Finally, we show that non-negative stable solutions of the full
H\'{e}non-Lane-Emden system are trivial provided \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}}).Comment: 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.htm