On the equation of a rotating film

We study positive periodic solutions to a nonautonomous nonlinear third-order ordinary differential equation of the theory of motion of a viscous incompressible fluid with free boundary. This equation describes the steady motions of a thin layer of a fluid film on the surface of a rotating horizontal cylinder in the gravity field. The linear operator on the left-hand side of the equation has a three-dimensional kernel. Moreover, the equation contains two nonnegative parameters proportional to the gravity acceleration and surface tension. Depending on these parameters the problem in question may have either two solutions or no solutions at all. We establish some qualitative properties of solutions to the problem: in particular, their asymptotic behavior at the extremal values of the parameters.