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}^α\left(|_{0}D_{t}^αu(t)|^{p-2}{_{0}}D_{t}^αu(t)\right) = f(t, u(t)), \;t\in [0,T],\\ &u(0) = u(T) = 0, \end{eqnarray*} where $0 < α<1$, $1<p<\infty$ 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 $0< α< \frac{1}{p}$ we obtain an almost every where classical solution.
math.AP↗