Viscosity solutions to the infinity Laplacian equation with lower terms

We establish the existence and uniqueness of viscosity solutions tothe Dirichlet problem $$\displaylines{ \Delta_\infty^h u=f(x,u), \quad \hbox{in } \Omega,\cr u=q, \quad\hbox{on }\partial\Omega,}$$ where \(q\in C(\partial\Omega)\), \(h>1\), \(\Delta_\infty^h u=|Du|^{h-3}\Delta_\infty u\). The operator \(\Delta_\infty u=\langle D^2uDu,Du \rangle\) is infinity Laplacian which strongly degenerate, quasilinear it associated with absolutely minimizing Lipschitz extension. When nonhomogeneous term \(f(x,t)\) non-decreasing in \(t\), we prove solution via Perron's method. also a result based on perturbation analysis solutions. If function nonpositive (nonnegative) non-increasing give by an iteration technique under condition that domain has small diameter. Furthermore, investigate to boundary-value singularity u=-b(x)g(u), \Omega, \cr u>0, u=0, \hbox{on }\partial\Omega, }$$ when satisfies some regular condition. analyze asymptotic estimates for near boundary.