Existence and symmetric result for Liouville–Weyl fractional nonlinear Schrödinger equation

•We introduce the principal fractional space.•We define the Nehari manifold and we prove some result considering this manifold.•We prove our two first results.•We consider some preliminaries for study the symmetry result.•We prove our final result.

We study the existence of positive solution for the one dimensional Schrödinger equation with mixed Liouville–Weyl fractional derivativestD∞α(-∞Dtαu(t))+V(t)u(t)=f(u(t)),t∈Ru∈Hα(R).Furthermore, we analyse radial symmetry property of these solutions. The proof is carried out by using variational methods jointly with comparison and rearrangement argument.