Ground State Solutions for a Class of Fractional Differential Equations with Dirichlet Boundary Value Condition

Abstract

In this paper, we apply the method of the Nehari manifold to study the fractional differential equation $(d/dt)((\mathrm{1}/\mathrm{2}){\hspace{0.17em}}_{\mathrm{0}}{D}_t^{-\beta }(u^{\prime }(t))+(\mathrm{1}/\mathrm{2}){\hspace{0.17em}}_t{D}_T^{-\beta }(u^{\prime }(t)))=f(t,u(t))$, a.e. $t\in [\mathrm{0},T],$ and $u\left(\mathrm{0}\right)=u\left(T\right)=\mathrm{0},$ where ${\hspace{0.17em}}_{\mathrm{0}}{D}_t^{-\beta },{\hspace{0.17em}}_t{D}_T^{-\beta }$ are the left and right Riemann-Liouville fractional integrals of order , respectively. We prove the existence of a ground state solution of the boundary value problem.