*Published Paper*

**Inserted:** 7 jul 2022

**Last Updated:** 7 jul 2022

**Journal:** Nonlinearity

**Volume:** 32

**Number:** 8

**Pages:** 3112-3143

**Year:** 2019

**Doi:** 10.1088/1361-6544/ab1273

**Abstract:**

The paper discusses nonlinear singular perturbations of delta type of the fractional Schr\"odinger equation $\imath\partial_t\psi=\left(-\triangle\right)^s\psi$, with $s\in(\frac{1}{2},1]$, in dimension one. Precisely, we investigate local and global well posedness (in a strong sense), conservations laws and existence of blow-up solutions and standing waves.