In micro-nano and even some macro-scale flows, the boundary slip cannot be ignored and has an important influence. We consider the most widely used Navier slip length model to investigate whether the solution of Stokes flow exists if a homogeneous slip is imposed as a boundary condition replacing the ordinary Dirichlet condition of velocity for some domains of simple geometries, including the two-dimensional channel, plane annulus and axisymmetric circular pipe. The Fourier series is employed to obtain the exact solutions formulated by the stream function. While a unique solution exists for most cases, it is found that yet occurs the cases of no solution under a certain slip length and infinitely multiple solutions under an infinitely discrete set of slip lengths, respectively.