I've been playing with an optimization problem and ended up reducing it to solving the following PDE for $h : \mathbb{R}^{d} \to \mathbb{R}^{d_1}$: $$ h = \nabla_x \Big[f\left(x, y+\nabla_y \big[g\left(x+h, y\right)\big]\right)\Big] $$ where $x \in \mathbb{R}^{d_1}$, $y \in \mathbb{R}^{d_2}$ with $d_1 + d_2 = d$ and $f,g : \mathbb{R}^{d} \to \mathbb{R}$ are fixed smooth functions. I have no particular boundary conditions.
I am no expert in PDEs and was wondering where to begin if I were to prove/disprove existence and uniqueness of solutions to this equation? I would be happy to begin with $d_1 = d_2 = 1$ for simplicity, which gives $$ h = \frac{\partial}{\partial x}\left[f\left(x, y+\frac{\partial}{\partial_y} \Big[g\left(x+h, y\right)\Big]\right)\right] \,. $$
