The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian

In this paper, we investigate a weighted Dirichlet eigenvalue problem for a class of degenerate operators related to the \begin{document}$ h $\end{document} degree homogeneous \begin{document}$ p $\end{document} -Laplacian\begin{document}$ \begin{equation} \nonumber \left \{ \begin{array}{ll} {|Du|^{h-1}}\Delta_p^N u+ \lambda a(x)|u|^{h-1}u = 0, \quad\quad \rm{in}\quad \Omega, \\ u = 0, \quad\quad \quad \quad \rm{on} \quad\partial\Omega. \end{array}\right. \end{equation} $\end{document}Here \begin{document}$ a(x) $\end{document} is a positive continuous bounded function in the closure of \begin{document}$ \Omega\subset \mathbb{R}^n(n\geq 2), $\end{document}\begin{document}$ h>1, $\end{document}\begin{document}$ 2< p<\infty, $\end{document} and \begin{document}$ \Delta_p^N u = \frac{1}{p}|Du|^{2-p} {\rm div}\left(|Du|^{p-2}Du\right) $\end{document} is the normalized version of the \begin{document}$ p $\end{document} -Laplacian arising from a stochastic game named Tug-of-War with noise. We prove the existence of the principal eigenvalue \begin{document}$ \lambda_\Omega $\end{document} , which is positive and has a corresponding positive eigenfunction for \begin{document}$ p>n $\end{document} . The method is based on the maximum principle and approach analysis to the weighted eigenvalue problem. When a parameter \begin{document}$ \lambda<\lambda_\Omega $\end{document} , we establish some existence and uniqueness results related to this problem. During this procedure, we also prove some regularity estimates including Hölder continuity and Harnack inequality.