A singular boundary value problem for the normalized <inline-formula><tex-math id="M1">$ p $</tex-math></inline-formula>-Laplacian equation

In this paper, we study the singular boundary value problem related to the normalized $ p $-Laplacian$ \begin{equation*} \left \{ \begin{array}{ll} \Delta_p^N u = \lambda f(x,u,Du) \quad &\text{in}\; \Omega, \\ u>0 \quad &\text{in}\; \Omega,\\ u = 0 \quad &\text{on} \;\partial\Omega, \end{array}\right. \end{equation*} $where $ \lambda>0 $ is a parameter and $ \Delta_p^N u = \frac{1}{p}|Du|^{2-p} {\rm div}\left(|Du|^{p-2}Du\right),\, 1<p<\infty $. The nonlinear term $ f(x,t,q):\Omega\times(0,\infty)\times\mathbb{R}^{n}\rightarrow \mathbb{R} $ is continuous and may exhibit singularity at $ t\rightarrow 0^{+} $. First, we establish the comparison principle by the perturbation method for the viscosity solution to the general equation $ \Delta_p^N u = F(x,u,Du) $ under some conditions on the term $ F(x,t,q) $. When $ 2\leq n<p<\infty $ and the parameter $ \lambda $ is lying in an appropriate range, we establish the existence of viscosity solutions to the singular boundary value problem in a bounded domain based on Perron's method and the comparison principle. Finally, we obtain the existence result in the whole Euclidean space by the approximation procedure.