Existence and symmetric result for Liouville-Weyl fractional nonlinear Schr\"odinger equation

We study the existence of positive solution for the one dimensional Schr"odinger equation with mixed Lioville-Weyl fractional derivatives \begin{eqnarray*}\label{Eq00} {t}D{\infty}^{\alpha}({{-\infty}}D{t}^{\alpha}u(t)) + V(t) u(t) = & f(u(t)),;;t\in $\mathbb{R}$\ u\in H^{\alpha}($\mathbb{R}$).\nonumber \end{eqnarray*} Furthermore, we analyse radial symmetry property of these solutions. The proof is carried out by using variational methods jointly with comparison and rearrangement argument.