The paper carried out an extensive comparative numerical analysis of the applicability of applied theoriesApplied theories of layered shells. A closed three-layer spherical shellThree-layer spherical shell was taken as the object of study. The outer layers of the shell are rigid and connected to each other by a lightweight filler (middle layer). The connection of the layers is assumed to be rigid. The three most applied theories in technical applications were taken for analysis. The first theory is based on the Kirchhoff—Love hypothesis for the entire package (single normal theory). The second one is based on the adoption of the Kirchhoff—Love hypotheses for each external loadExternal load-bearing layer, and for the filler—constancy in shear thickness. This theory is usually called the broken element theory in scientific literature. The third applied theory generalizes the first theory by considering transverse shear deformationsDeformation. First, an axisymmetric longitude boundary problem for an inhomogeneous spherical shell is solved analytically. On the outer boundary surface of the shell, the load is taken to be axisymmetric in longitude, and in latitude it is determined by the Legendre polynomial of the n-th order, which determines its sufficient variability along the meridian from weak to strong. The spatial exact solution of this problem was obtained by the matrix method. Then, based on the asymptotic integration of 3D equations of the linear theory of elasticity, a solution asymptotic in terms of the shell thinness parameter was constructed.