Fully nonlinear degenerate equations with applications to Grad equations

We consider a class of degenerate elliptic fully nonlinear equations with applications to Grad equations: \begin{align} \begin{cases} |Du|^\gamma \mathcal{M}_{\lambda,\Lambda}^+\big(D^2u(x)\big)=f\big(|u\geq u(x)|\big) &\text{ in }\Omega, u=g &\text{ on }\partial\Omega, \end{cases} \end{align} where $\gamma\geq 1$ is a constant, $\Omega$ is a bounded domain in $\mathbb{R}^N$ with $C^{1,1}$ boundary. We prove the existence of a $W^{2,p}$-viscosity solution to the above equation, which degenerates when the gradient of the solution vanishes.