Existence of Solutions for some Quasiaffine PDEs

We prove the existence of solutions of problems of the type \begin{equation*} \left\{ \begin{aligned} \Phi(Du(x)) &= f(x) &\quad &\text{\rm in } \, \Omega\\ u(x) &=\xi_0 x &\quad &\text{\rm on } \, \partial \Omega \end{aligned} \right. \end{equation*} with $\Phi:\mathbb{R}^{n\times n} \to \mathbb{R}$ quasiaffine function and $\xi_0\in \mathbb{R}^{n\times n}$ fixed.