On the Hénon-Lane-Emden conjecture

Details On the Hénon-Lane-Emden conjecture On the Hénon-Lane-Emden conjecture

2011-07-28

arxiv.org/abs/1107.5611v1

Mathematics

Analysis of PDEs

20 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/

Scientific paper

We consider the problem of non-existence of solutions for the following H\'{e}non-Lane-Emden system {eqnarray*} \{{array}{lcl} \hfill -\Delta u&=& |x|^{a}v^p \ \ \text{in} \mathbb{R}^N, \hfill -\Delta v&=& |x|^{b}u^q \text{in}\mathbb{R}^N, {array}. {eqnarray*} when $pq>1$, $p,q,a,b\ge0$, and $(p,q)$ are {\it under} the critical hyperbola, i.e. $ \frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that there is no positive bounded solution in dimension N=3, extending a result established recently by Phan-Souplet in the scalar case. This solves the {\it H\'{e}non-Lane-Emden conjecture} in dimension N=3 for bounded positive solutions. For the scalar cases, whether of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $N\ge 3$ (resp., $N\ge 5$), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1

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