Global uniqueness for the fractional semilinear Schrödinger equation

Abstract

We study global uniqueness in an inverse problem for the fractional semilinear Schrödinger equation $(-\Delta )^{s}u+q(x,u)=0$ with $s\in (0,1)$. We show that an unknown function $q(x,u)$ 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 an $L^\infty$ estimate for this nonlocal equation under appropriate regularity assumptions.