Existence and multiplicity result for a fractional p-Laplacian equation with combined fractional derivatives

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.