Ground state solutions for a class of fractional differential equations with Dirichlet boundary value condition. (English) Zbl 1474.34034

Hu, Zhigang; Liu, Wenbin; Liu, Jiaying

Abstr. Appl. Anal. 2014, Article ID 958420, 7 p. (2014).

Summary: In this paper, we apply the method of the Nehari manifold to study the fractional differential equation \((d / d t)((1 / 2) {}_0 D_t^{- \beta}(u'(t)) +(1 / 2) {}_t D_T^{- \beta}(u'(t))) = f(t, u(t))\), a.e. \(t \in [0, T]\), and \(u \left(0\right) = u \left(T\right) = 0\), where \({}_0 D_t^{- \beta}\), \({}_t D_T^{- \beta}\) are the left and right Riemann-Liouville fractional integrals of order \(0 \leq \beta < 1\), respectively. We prove the existence of a ground state solution of the boundary value problem.

MSC:

34A08	Fractional ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations

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