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NEHARI MANIFOLD AND MULTIPLICITY RESULTS FOR A CLASS OF FRACTIONAL BOUNDARY VALUE PROBLEMS WITH p-LAPLACIAN

  • Ghanmi, Abdeljabbar (Department of Mathematics Faculty of Science and Arts Khulais University of Jeddah) ;
  • Zhang, Ziheng (School of Mathematical Sciences Tianjin Polytechnic University)
  • Received : 2018.12.02
  • Accepted : 2019.04.25
  • Published : 2019.09.30

Abstract

In this work, we investigate the following fractional boundary value problems $$\{_tD^{\alpha}_T({\mid}_0D^{\alpha}_t(u(t)){\mid}^{p-2}_0D^{\alpha}_tu(t))\\={\nabla}W(t,u(t))+{\lambda}g(t){\mid}u(t){\mid}^{q-2}u(t),\;t{\in}(0,T),\\u(0)=u(T)=0,$$ 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

Acknowledgement

Supported by : NSFC

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