Nehari manifold and multiplicity results for a class of fractional boundary value problems with $p$-Laplacian
Bull. Korean Math. Soc. 2019 Vol. 56, No. 5, 1297-1314
https://doi.org/10.4134/BKMS.b181172
Published online September 30, 2019
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
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