Summary: The aim of this paper is to obtain the existence of solution for the fractional \(p\)-Laplacian Dirichlet problem with mixed derivatives \[ \begin{aligned} _tD_T^\alpha(|_0D_t^\alpha u(t)|^{p-2}_0D_t^\alpha u(t)) = f(t,u(t)), \quad t\in [0,T],\\ u(0) = u(T) = 0,\end{aligned} \] where \(\frac{1}{p} <\alpha < 1\), \(1 < p < \infty\) and \(f : [0,T] \times \mathbb{R}\to \mathbb{R}\) 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.