I'd like to know whether there has been work on Monge-Ampere like equations of the form $$\det u'(x)=f(x,u(x)),$$ where $u:\mathbb{R}^n\to \mathbb{R}^n$ is $C^1$ and $f$ is a positive function.
Under the additional assumption $u=\nabla v(x)$, this becomes a special case of the Monge-Ampere equation.
What is known about uniqueness and/or existence for appropriate boundary conditions?
