We consider positive solutions of the Dirichlet problem for the fractional Laplace equation with singular nonlinearity ( - Δ ) s u ( x ) = K ( x ) u - α ( x ) + μ u p - 1 ( x ) in Ω , u > 0 in Ω , u = 0 in Ω c : = R N \ Ω , where s ∈ ( 0 , 1 ) , α > 0 and Ω ⊂ R N is a bounded domain with smooth boundary ∂ Ω and N > 2 s . Under some appropriate assumptions of α , p , μ and K , we obtain the existence of multiple weak solutions, and among them, including the minimal solution and a ground state solution. Radial symmetry of C loc 1 , 1 ∩ L ∞ solutions are also established for subcritical exponent p when the domain is a ball. Nonexistence of C 1 , 1 ∩ L ∞ solutions are obtained for star-shaped domain under a condition of K .