We study the stability of a three-layer shell with light filler. The shell is supported by the system of longitudinal ribs. The inner surface of the shell is attached to a hollow elastic cylinder. We consider the shell under an axial compression. Using the representation of the displacement functions for the shell and the cylinder in terms of double trigonometric series, we transform the problem to a problem for the system of three homogeneous algebraic equations. Equating the determinant of the last system of equations to the zero, we obtain the characteristic equation for the determining critical parameters.