We consider the periodic fractional nonlinear Schrödinger equation \(\begin{aligned} iu_t -(-\Delta )^{\frac{s}{2}} u + {\mathcal {N}}(|u|)u=0, \quad x\in {\mathbb {T}}^N,\, \, t \in \mathbb R, \, \, s>0, \end{aligned}\) where the nonlinearity term is expressed in two ways: the first one \({\mathcal {N}}\in C^J(\mathbb R^+)\) , whose derivatives have a certain polynomial decay, e.g., \({\mathcal {N}}(|u|)=\log (|u|)\) ; the second one is given by a sum of powers, possibly infinite, \(\begin{aligned} {\mathcal {N}}(|u|) = \sum a_k |u|^{\gamma _k}, \quad \gamma _k \in {\mathbb {R}}, ~~ a_k \in {\mathbb {C}}, \end{aligned}\) which includes examples such as \({\mathcal {N}}(|u|) \, u =\frac{u}{|u|^{\gamma }},\) \(\gamma >0\) . By using standard properties of periodic Sobolev spaces \(H^J({\mathbb {T}}^N)\) , \(J>0\) , we study the local well-posedness for the Cauchy problems of the above equations when initial data satisfies a non-vanishing condition \(\inf \limits _{x\in {\mathbb {T}}^N}|u_0(x)|>0\) .
