A new attempt

This is an attempt to incorporate the density using a slightly different reasoning.

Let F be the total stationary heat flux. Each segment of the atmosphere will not absorb the total flux, however, but we assume that the absorbed flux is proportional to the mass density. Assuming an exponential decay in density the heat equation becomes:

where m is a phenomenological parameter proportional to but not identical to the atmospheric mass. The solution is of course:

which smoothly approaches a zero lapse rate, but where the boundary condition still needs to be specified.