  This paper is concerned with the forward and inverse problems for the
fractional semilinear elliptic equation $(-\Delta)^s u +a(x,u)=0$ for $0<s<1$.
For the forward problem, we proved the problem is well-posed and has a unique
solution for small exterior data. The inverse problems we consider here
consists of two cases. First we demonstrate that an unknown coefficient
$a(x,u)$ can be uniquely determined from the knowledge of exterior
measurements, known as the Dirichlet-to-Neumann map. Second, despite the
presence of an unknown obstacle in the media, we show that the obstacle and the
coefficient can be recovered concurrently from these measurements. Finally, we
investigate that these two fractional inverse problems can also be solved by
using a single measurement, and all results hold for any dimension $n\geq 1$.
