A weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian

Abstract In this paper, we study a weighted eigenvalue problem of the degenerate operator associated with infinity Laplacian Δ ∞ h u + λ a ( x ) | u | h − 1 u = 0 , in Ω , u = 0 , on ∂ Ω , where h > 1 , Δ ∞ h u = | D u | h − 3 Δ ∞ u is the h -homogeneous infinity Laplacian and a ( x ) is a positive continuous bounded function in Ω . 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.