A note on existence of smooth solution to Jacobian equation for compactly supported smooth data in $\mathbb{R}^2$

In this note, we give an explicit construction of global solutions to the prescribed Jacobian equation \[ \det(\nabla u)=f\mbox{ in }\mathbb{R}^2, \] for a class of data. For every $f\in C_c^1(\mathbb{R}^2)$ and $p>1$, we construct a solution $u\in \dot W^{1,2p}(\mathbb{R}^2)\cap L^\infty(\m