# Hamiltonian-type elliptic system

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 $\nabla G=(G_{u},G_{v})$.

Title Hamiltonian-type elliptic system HamiltoniantypeEllipticSystem 2013-03-11 19:28:32 2013-03-11 19:28:32 linor (11198) (0) 1 linor (0) Definition