  The aim of this paper is to obtain the existence of solutions for the
following fractional p-Laplacian Dirichlet problem with mixed derivatives
\begin{eqnarray*}
&{_{t}}D_{T}^{\alpha}\left(|_{0}D_{t}^{\alpha}u(t)|^{p-2}{_{0}}D_{t}^{\alpha}u(t)\right)
= f(t, u(t)), \;t\in [0,T],\\ &u(0) = u(T) = 0, \end{eqnarray*} where $0 <
\alpha <1$, $1<p<\infty$ and $f:[0,T]\times \mathbb{R} \to \mathbb{R}$ is a
continuous function. We obtain the existence of nontrivial solutions by using
the direct method in variational methods and the genus in the critical point
theory. Furthermore, if $0< \alpha < \frac{1}{p}$ we obtain an almost every
where classical solution.
