The Nehari manifold for a boundary value problem involving Riemann–Liouville fractional derivative

Abstract

We aim to investigate the following nonlinear boundary value problems of fractional differential equations: ( P λ ) { − t D 1 α ( | 0 D t α ( u ( t ) ) | p − 2 0 D t α u ( t ) ) = f ( t , u ( t ) ) + λ g ( t ) | u ( t ) | q − 2 u ( t ) ( t ∈ ( 0 , 1 ) ) , u ( 0 ) = u ( 1 ) = 0 , $$\begin{aligned} (\mathrm{P}_{\lambda}) \left \{ \textstyle\begin{array}{l} -_{t}D_{1}^{\alpha} ( \vert {}_{0}D_{t}^{\alpha}(u(t)) \vert ^{p-2} {}_{0}D_{t}^{\alpha}u(t) ) \\ \quad=f(t,u(t))+\lambda g(t) \vert u(t) \vert ^{q-2}u(t)\quad (t\in(0,1)),\\ u(0)=u(1)=0, \end{array}\displaystyle \right . \end{aligned}$$ where λ is a positive parameter, 2 < r < p < q $2< r< p< q$ , 1 2 < α < 1 $\frac{1}{2}<\alpha < 1$ , g ∈ C ( [ 0 , 1 ] ) $g\in C([0,1])$ , and f ∈ C ( [ 0 , 1 ] × R , R ) $f\in C([0,1]\times\mathbb{R},\mathbb{R})$ . Under appropriate assumptions on the function f, we employ the method of Nehari manifold combined with the fibering maps in order to show the existence of solutions to the boundary value problem for the nonlinear fractional differential equations with Riemann–Liouville fractional derivative. We also present an example as an application.