LaneEmden System
Here is an example of Hamiltonian elliptic systems,
also called the LaneEmden system,

$$(LE)\{\begin{array}{cc}\mathrm{\Delta}u={v}^{p}\hfill & x\in \mathrm{\Omega},\hfill \\ \mathrm{\Delta}v={u}^{q}\hfill & x\in \mathrm{\Omega},\hfill \\ u=v=0\hfill & x\in \partial \mathrm{\Omega},\hfill \end{array}$$ 

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

$$J(u,v)={\int}_{\mathrm{\Omega}}\nabla u\nabla vdx{\int}_{\mathrm{\Omega}}(\frac{1}{q+1}{u}^{q+1}+\frac{1}{p+1}{v}^{p+1})\mathit{d}x.$$ 
