The Nehari Manifold and its Application to a Fractional Boundary Value Problem

Abstract

In this paper, we study the Nehari manifold and its application to the following fractional boundary value problem: $$\begin{aligned} {\left\{ \begin{array}{ll} - \frac{d}{d t} \Big (\frac{1}{2} {}_0D_t^{- \beta } (u^{\prime } (t)) + \frac{1}{2} {}_tD_T^{- \beta } (u^{\prime } (t)) \Big )\\ \quad = \lambda f (t) (u (t))^{p - 1} + g (t) (u(t))^{q - 1}, &{} \mathrm{a.e.}\;\; t \in [0, T],\\ u (0) = u (T) = 0, \end{array}\right. } \end{aligned}$$ where \({}_0D_t^{- \beta }\) and \({}_tD_T^{- \beta }\) are the left and right Riemann–Liouville fractional integrals of order \(0 \le \beta < 1\), respectively. We prove that the problem has at least two nontrivial non-negative solutions when the parameter \(\lambda \) belongs to a certain subset of \(\mathbb{R }\).