In this talk, I will introduce the lower bound of the first Dirichlet eigenvalue of bounded domains for the p-Laplacian under the condition that the domain admits a specific function which fulfills certain criteria related to divergence and gradient conditions. As an application, we provide an estimation for the first Dirichlet eigenvalue of geodesic balls with large radius in asymptotically hyperbolic Einstein manifolds. We also obtain the estimate in non-compact manifold with nonnegative Ricci curvature. Part of the work is joint with Zhiwei Lv.