Global uniqueness for the semilinear fractional Schr\"odinger equation

We study global uniqueness in an inverse problem for the fractional semilinear Schr"{o}dinger equation $(-\mathrm{\Delta }{)}^{s}u+q(x,u)=0$ with $s\in (0,1)$\mathrm{.}Weshowthatanunknownfunction$q(x,u)$ can be uniquely determined by the Cauchy data set. In particular, this result holds for any space dimension greater than or equal to $2$. Moreover, we demonstrate the comparison principle and provide a $L^{\mathrm{\infty }}$ estimate for this nonlocal equation under appropriate regularity assumptions.