I am solving elliptic PDE problem, for which, Euler scheme looks as following: $$ \nabla [\gamma ( |\nabla u|^2) \nabla u] = 0,$$ where $$\gamma(|\nabla u|^2) = (1 + |\nabla u|^2)^{-1/2}. $$
I am aiming to define minimization problem for problem above. Does anybody have any suggestion how it should look like? Or some recommended literature?
For the particular equation you are solving (called the minimal surface equation), the functional you are trying to minimize is $$ J(u) = \int_\Omega \sqrt{1 + |\nabla u|^2} \; dx. $$
You can find a derivation of the equations, as well as a discussion of solution approaches in lectures 31.5 and following here: http://www.math.tamu.edu/~bangerth/videos.html . Of course there are many other sources for the same material as well. For example, nearly every book about the calculus of variations will have your example.
Let's assume, that you have discretized problem and on every point you obtained evaluation $ g_{ij} $. Therefore $g(u) = 0$ is your system of nonlinear equations.
Now let $\gamma$ to be norm function defined as $\gamma(x) = || g(x)||^2$.
Then your nonlinear eq. problem is equivalent to the problem $$ \text{minimize} \ \ \gamma(x) .$$
