Ground state solution for a class of differential equations with left and right fractional derivatives

Abstract: In this work we study the existence of solution for a class of fractional differential equation given by \begin{eqnarray}\label{eq00} {t}D{\infty}^{\alpha}{{-\infty}}D{t}^{\alpha}u(t) + u(t) = & f(t,u(t))\ u\in H^{\alpha}(\mathbb{R}).\nonumber \end{eqnarray} where $\alpha \in (1/2, 1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}$, $f\in C(\mathbb{R}, \mathbb{R})$. Using mountain pass theorem and comparison argument we prove that (\ref{eq00}) at least has one nontrivial solution.