Boundary value problem with fractional p-Laplacian operator

TL;DR

This paper establishes the existence of solutions for a boundary value problem involving a fractional p-Laplacian operator with mixed derivatives, using variational methods and the Mountain Pass Theorem.

Contribution

It introduces a new existence result for fractional p-Laplacian boundary value problems with mixed derivatives under growth conditions on the nonlinearity.

Findings

Existence of nontrivial solutions proven.

Application of Mountain Pass Theorem to fractional operators.

Results extend known solutions to fractional p-Laplacian problems.

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.