40A05-LaneEmdenSystem3.tex

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\pmtitle{Lane-Emden System}
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Here is an example of Hamiltonian elliptic systems,
also called the Lane-Emden system,
%\be  %\label{LE}
$$
(LE) \; \left\{
\begin{array}{ll}
 - \Delta u =  v^p & \, x \in\Omega,\nonumber\\
 - \Delta v =  u^q & \, x \in\Omega,\nonumber\\
             u=v=0 & \,  x \in \partial \Omega, \nonumber
\end{array}
\right.
%J(u)=\int_{\Omega}\Big[\frac{1}{2}|\nabla %u|^2-\frac{1}{q+1}|x|^r|u|^{q+1}\Big]dx
%\ee
$$
where $p,q>0,\Omega\subset {\mathbb R}^N (N\ge 1)$ 
is an open bounded domain. (LE) is called sublinear (superlinear) 
if $pq<1 \; (pq>1)$.
The associated energy functional to (LE) is 
$$
J(u,v)=\int_{\Omega}\nabla u  \nabla v dx -
\int_{\Omega}(\frac{1}{q+1}u^{q+1}+\frac{1}{p+1}v^{p+1})dx.
$$

%Here are some numerical results to the Liouville-Gelfand problem
%when $\Omega$ is an annulus.

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