# Lane-Emden System

Lane-Emden System Here is an example of Hamiltonian elliptic systems, also called the Lane-Emden system,

 $(LE)\;\left\{\begin{array}[]{ll}-\Delta u=v^{p}&\,x\in\Omega,\\ -\Delta v=u^{q}&\,x\in\Omega,\\ u=v=0&\,x\in\partial\Omega,\end{array}\right.$

where $p,q>0,\Omega\subset{\mathbb{R}}^{N}(N\geq 1)$ is an open bounded domain. (LE) is called sublinear (superlinear) if $pq<1\;(pq>1)$. The associated energy functional to (LE) is

 $J(u,v)=\int_{\Omega}\nabla u\nabla vdx-\int_{\Omega}(\frac{1}{q+1}u^{q+1}+% \frac{1}{p+1}v^{p+1})dx.$
Title Lane-Emden System LaneEmdenSystem 2013-03-11 19:34:47 2013-03-11 19:34:47 linor (11198) (0) 1 linor (0) Definition