In my research problem, I'm arrived at the following simple looking but highly non-linear pde which is related to the von Karman equations for plates with incompatible elastic strain (http://rspa.royalsocietypublishing.org/content/467/2126/402).

A sufficiently smooth (possibly analytic) function $w:X\to\mathbb{R}$ is given where $X$ is a simply connected bounded set in $\mathbb{R}^2$. Consider the partial differential equation

$$[\zeta,\zeta]=2[\zeta,w]$$

where $[f,g]:=f_{,xx} g_{,yy} + f_{,yy} g_{,xx} - 2 f_{,xy} g_{,xy}$.

Is anything known about the solution $\zeta:X\to\mathbb{R}$, in any category, of the above equation? Any reference would be appreciated!