# Nonvariational systems

Nonvariational systems

A typical nonvariational elliptic system has the form

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

where $\Omega\subset{\mathbb{R}}^{N}(N\geq 1)$ is an open bounded domain, $f(x;u,v),g(x;u,v)\in\mathcal{C}^{1}(\overline{\Omega}\times\mathbb{R}^{2};% \mathbb{R})$ in the variables $(u,v)\in\mathbb{R}^{2}$. Here, we further assume that there exists no function $G(x;u,v)$ with $\nabla G=(f,\pm g)$ or $\nabla G=(g,f)$. Under this assumption, it is easy to see that problem (NV) is nonvariational.

Title Nonvariational systems NonvariationalSystems1 2013-03-11 19:28:55 2013-03-11 19:28:55 linor (11198) (0) 1 linor (0) Definition