Lane-Emden System

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

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

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