This is not really an answer, but I prefer the luxury of the answer box instead of the rather spartan comment box.

You have only one equation for $n$ unknown functions (the components of the vector-valued function $u$), so the equation is underdetermined. This gives you a lot of flexibility on what to do. Roughly speaking, you get to impose $n-1$ additional conditions in order to get a well-posed boundary value problem for a system of PDE's.

There might be a clever choice of the $n-1$ conditions that makes your question very easy to answer, but I don't know and haven't been able to think of any.

The most common way to do this is to set $u$ equal to the gradient of a scalar function $\phi$, which turns your equation into a real Monge-Ampere equation. If the function $f$ were always positive, then you can try to solve for $\phi$ convex. There is a well-developed theory of this situation, because the PDE becomes second order elliptic and many techniques are available.

However, if $f$ changes sign, then much, much less is known. But here are some partial answers:

1) If you restrict to $n=2$, assume $\Omega$ is a nice smooth domain, and assume that $f$ changes sign "cleanly" (i.e., $f$ vanishes along a smooth curve in $\Omega$ and its gradient is everywhere nonzero along this curve), then the equation becomes what is known as a nonlinear Tricomi equation, which has been studied. You can probably set up some appropriate boundary value problem to solve your equation on $\Omega$ this way.

2) The ideas in 1) probably could be generalized to higher dimensions, but I'm unaware of any prior work on this. The idea is to solve for $\phi$ to have positive definite Hessian where $f$ is positive and signature $(n-1,1)$ where $f$ is negative. Then the PDE becomes elliptic where $f$ is positive and hyperbolic where $f$ is negative.

3) All of this might be able to be recast in the form of a nonlinear symmetric positive system as defined and studied by K. O. Friederichs.

4) Maybe the most promising but still rather difficult direction is to *not* use the Monge-Ampere equation but try to find a different set of $n-1$ additional first order equations to impose on $u$, (so you get an $n$-by-$n$ system of first order PDE's) such that the resulting system is symmetric positive. Then you try to identify the right boundary conditions that both leads to the existence of a solution and implies that $u$ vanishes on the boundary of $\Omega$.

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