摘要：In this paper, the author considers the following nonlinear fractional boundary value problem: { d d t ( 1 2 0 D t − β ( u ′ ( t ) ) + 1 2 t D T − β ( u ′ ( t ) ) ) + ∇ F ( t , u ( t ) ) = 0 , a.e. t ∈ [ 0 , T ] , u ( 0 ) = u ( T ) = 0 , $$\left \{ \textstyle\begin{array}{l} \frac{d}{dt} ( {\frac),{ }_(D_{t}^{-\beta} ({u}'(t))+\frac),{ }_{t}D_{T}^{-\beta} ({u}'(t))} )+\nabla F(t,u(t))=0, \quad\mbox{a.e. } t\in[0,T], \\ u(0)=u(T)=0, \end{array}\displaystyle \right . $$ where D t − β 0 ${ }_(D_{t}^{-\beta} $ and D T − β t ${ }_{t}D_{T}^{-\beta} $ are the left and right Riemann-Liouville fractional integrals of order 0 ≤ β < 1 $0\le\beta<1$ , respectively, ∇ F ( t , x ) $\nabla F(t,x)$ is the gradient of F at x. By applying the variant fountain theorems, the author obtains the existence of infinitely many small or high energy solutions to the above boundary value problem.
关键词：fractional differential equations ; variant fountain theorems ; critical point theory ; variational method
