In this paper, we study a weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian $\left\{\begin{array}{c}{\Delta }_{\infty }^{h}u+\lambda a\left(x\right){\left|u\right|}^{h-1}u=0,\mathrm{in}\Omega ,\hfill \\ u=0,\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$where $h>1,$ ${\Delta }_{\infty }^{h}u={\left|Du\right|}^{h-3}{\Delta }_{\infty }u$ is the $h$-homogeneous infinity Laplacian and $a\left(x\right)$ is a positive continuous bounded function in $\Omega .$ We prove the existence of the principal eigenvalue and a corresponding positive eigenfunction. The approach to the weighted eigenvalue problem is based on the maximum principle and when a parameter is less than the principal eigenvalue, some existence and uniqueness results related to this problem are established.

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与无穷拉普拉斯算子相关的退化算子的加权特征值问题

在本文中，我们研究了与无穷拉普拉斯算子相关联的退化算子的加权特征值问题 $\left\{\begin{array}{c}{\Delta }_{\infty }^Hü+\lambda \mathrm{一种}（X）{\left|ü\right|}^{H-\mathrm{1个}}ü=0，\mathrm{在}\Omega ，\hfill \\ ü=0，\mathrm{上}\partial \Omega ，\hfill \end{array}\right.$哪里 $H>\mathrm{1个}，$ ${\Delta }_{\infty }^Hü={\left|dü\right|}^{H-3}{\Delta }_{\infty }ü$ 是个 $H$-齐次无限拉普拉斯算子和 $\mathrm{一种}（X）$ 是一个正连续有界函数 $\Omega 。$我们证明了主特征值和相应的正特征函数的存在。加权特征值问题的方法基于最大原理，当参数小于主要特征值时，就建立了与此问题相关的一些存在性和唯一性结果。