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

TL;DR

This paper proves the existence of solutions for a fractional p-Laplacian boundary value problem with mixed derivatives, using variational methods and critical point theory, and establishes conditions for classical solutions.

Contribution

It introduces new existence results for fractional p-Laplacian equations with combined derivatives, employing variational and critical point techniques.

Findings

Existence of nontrivial solutions under certain conditions.

Classical solutions exist almost everywhere when 0<α<1/p.

Use of variational methods and genus in critical point theory.

Abstract

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.