This talk is concerned with the Dirichlet eigenvalue problem associated to the infinity Laplacian in metric spaces. We provide a direct PDE approach to find the principal eigenvalue and eigenfunctions for a bounded domain in a proper geodesic space with no measure structure. We give an appropriate notion of solutions to the infinity eigenvalue problem and show the existence of solutions by adapting Perron's method. Our method is different from the standard limit process, introduced by Juutinen, Lindqvist and Manfredi in 1999, via the variational eigenvalue formulation for p-Laplacian in the Euclidean space. This talk is based on joint work with Ayato Mitsuishi at Fukuoka University.

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