Multiple solutions for a nonlinear fractional boundary value problem via critical point theory
Summary: This paper is concerned with the existence of multiple solutions for the following nonlinear fractional boundary value problem: \[ D_{T^-}^\alpha (a(x) D_{0^+}^\alpha u(x)) = f(x, u(x)), \quad x \in [0, T], \quad u(0) = u (T) = 0, \] where \(\alpha \in (1/2, 1]\), \(a(x)\in L^{\infty}[0, T]\) with \(a_0 = \operatorname{ess}\inf_{x\in [0,T]} a(x) > 0\), \(D_{T^-}^\alpha\) and \(D_{0^+}^\alpha\) stand for the left and right Riemann-Liouville fractional derivatives of order \(\alpha\), respectively, and \(f:[0,T] \times \mathbb{R} \rightarrow \mathbb{R}\) is continuous. The existence of infinitely many nontrivial high or small energy solutions is obtained by using variant fountain theorems.
Riemann-Liouville fractional derivatives
scientific article; zbMATH DE number 6816050