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 $\abla G=(f,\\pm g)$ or $\abla G=(g,f)$ . Under this assumption,it is easy to see that problem (NV) is nonvariational.