  We prove a fractional Pohozaev type identity in a generalized framework and
discuss its applications. Specifically, we shall consider applications to
nonexistence of solutions in the case of supercritical semilinear Dirichlet
problems and regarding a Hadamard formula for the derivative of Dirichlet
eigenvalues of the fractional Laplacian with respect to domain deformations. We
also derive the simplicity of radial eigenvalues in the case of radial bounded
domains and apply the Hadamard formula to this case.
