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|^γ$\mathcal{M}$_{λ,Λ}^+\big(D^2u(x)\big)=f\big(|u\geq u(x)|\big) &$\text{ in }$Ω, u=g &$\text{ on }$\partialΩ, \end{cases} \end{align} where $\gamma \ge 1$ is a constant, $\mathrm{\Omega }$ 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.