The growth of aerosol particles by homogeneous condensation in the presence of a spatially uniform source of a condensing monocomponent vapor is investigated. The kinetic model in which the condensation growth obeys the set of Formal kinetic equations describing the process (g) + (1) --> (g + 1) (with g being the number of vapor molecules in a particle) is used. The mass spectrum is expressed in terms of the function describing time evolution of the vapor concentration. In contrast to the case of free condensation (no external source), the mass spectrum is a smooth function of the particle mass with the frontal peak moving with time to the right along the mass axis. We have shown (Lushnikov and Kulmala, 1995, Phys. Rev. E 52, 1658-1668) that the particle number concentration either grows unlimitedly with increasing time or remains finite depending on how fast the condensation coefficients alpha(g) describing the rates for monomer accretion grow with the particle mass g. For power dependencies alpha(g) proportional to g(lambda) the condensation regime switches at lambda = 1/2. The analogy of this phenomenon to the second-order phase transitions is noted. The present numerical results illustrate the growth processes using condensation coefficients obtained from Fuchs and Sutugin (1970, J. Colloid Interface Sci. 27, 216-222). The analysis of transition regime shows that the particle mass spectrum can be expressed in terms of a universal function whereas the dependence on the Knudsen number enters in a rather trivial manner in the final expression for the mass spectrum.