The aim of this paper is to obtain the existence of solution for the fractional p-Laplacian Dirichlet problem with mixed derivatives (Equation Presented) where 1/p < α < 1, 1 < p < ∞ and f : [0,T] × ℝ → ℝ is a Carathéodory function which satisfies some growth conditions. We obtain the existence of nontrivial solutions by using the direct method in variational methods and mountain pass theorem.