Nehari Manifold and Multiplicity Results for a Class of Fractional Boundary Value Problems with $p$-Laplacian

Bull. Korean Math. Soc. Published online May 29, 2019

Abdeljabbar Ghanmi and Ziheng Zhang
University of Jeddah, Tianjin Polytechnic University

Abstract : In this work, we investigate the following fractional boundary value problems
\begin{eqnarray*}
\left\{\begin{array}{ll}
_{t}D_{T}^{\alpha}\left(|_{0}D_{t}^{\alpha}(u(t))|^{p-2} {_0 D}_{t}^{\alpha}u(t)\right)=\nabla W(t,u(t))+\lambda g(t) |u(t)|^{q-2}u(t),\;t\in (0,T),\\[0.25cm]
u(0)=u(T)=0,
\end{array}
\right.
\end{eqnarray*}
where $\nabla W(t,u)$ is the gradient of $W(t,u)$ at $u$ and $W\in C([0,T]\times \mathbb{R}^{n},\mathbb{R})$ is homogeneous of degree $r$, $\lambda$ is a positive parameter, $g\in C([0,T])$, $1<r<p<q$ and $\frac{1}{p}<\alpha<1$. Using the Fibering map and Nehari manifold, for some positive constant $\lambda_0$ such that
$0<\lambda<\lambda_0$, we prove the existence of at least two non-trivial solutions.

Keywords : Nonlinear fractional differential equations; Boundary value problem; Existence of solutions; Nehari Manifold