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.