On Fractional Schrödinger Equations in sobolev spaces

This is the authors' abstract. We don't add key points for this paper.

Let σ ∈ (0,1) with σ ≠ 1/2. We investigate the fractional nonlinear Schrödinger equation in Rᵈ:{eqnarray}i∂_tu+(-Δ)^σ u+μ|u|ᵖ⁻¹u=0, u(0)=u_0∈ H^s,{eqnarray}where (-Δ)^σ is the Fourier multiplier of symbol |ξ|2σ, and μ=± 1. This model has been introduced by Laskin in quantum physics [23]. We establish local well-posedness and ill-posedness in Sobolev spaces for power-type nonlinearities.