Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2019; 56(5): 1297-1314

Online first article August 6, 2019      Printed September 30, 2019

https://doi.org/10.4134/BKMS.b181172

Copyright © The Korean Mathematical Society.

Nehari manifold and multiplicity results for a class of fractional boundary value problems with $p$-Laplacian

Abdeljabbar Ghanmi, Ziheng Zhang

University of Jeddah, KSA; 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),\\[0.2em] 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

Keywords: nonlinear fractional differential equations, boundary value problem, existence of solutions, Nehari manifold

MSC numbers: 34A08, 34A12, 35B15

Supported by: The second author was partially supported by the NSFC (11771044, 61503279)