Hamiltonian-type elliptic system

A hamiltonian-type elliptic system is of the form

(HS)\\;\\;\\left\\{\\begin{array}[]{ll}-\\Delta u=G_{v}(x;u,v),&\\,x\\in\\Omega\\\\-\\Delta v=G_{u}(x;u,v),&\\,x\\in\\Omega\\\\u=v=0\\;\\mbox{or}\\;\\frac{\\partial u}{\\partial n}=\\frac{\\partial v}{\\partial n}=%0&\\,x\\in\\partial\\Omega\\end{array}\\right.

where $\\Omega\\subset{\\mathbb{R}}^{N}(N\\geq 1)$ is an open bounded domain, $G(x;u,v)\\in\\mathcal{C}^{1}(\\overline{\\Omega}\\times\\mathbb{R}^{2};\\mathbb{R})$ in the variables $(u,v)\\in\\mathbb{R}^{2}$ with $\abla G=(G_{u},G_{v})$ .