A general perturbation theory and its integral equation formulation are presented in the context of density functional theory. The integral equation approach can be used to generate potentials yielding very accurate approximate ground-state densities and orbital energies within the framework of Kohn-Sham theory. This formulation eliminates the necessity of directly taking cumbersome functional derivatives in order to arrive at the components of the unknown Kohn-Sham potential, including the correlation potential. The asymptotic properties of the second-order correlation potentials are investigated. Consistent with the work by Facco Bonetti [Phys. Rev. Lett. 86, 2241 (2001)], it is shown that the second-order potentials are divergent, and the source of the incorrect long range decay is identified. An approximate second-order potential with corrected long range properties is proposed.