Boundary value problem with fractional p-Laplacian operator

Abstract

The aim of this paper is to obtain the existence of solution for the 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 $\frac{1}{p} < \alpha <1$, $1<p<\infty$ and $f:[0,T]\times \mathbb{R} \to \mathbb{R}$ is a Carath\'eodory function wich satisfies some growth conditions. We obtain the existence of nontrivial solution by using the Mountain Pass Theorem.