Bifurcation of axisymmetric equilibrium states of inhomogeneous annular and circular plates subjected to internal pressure is studied. The effect of material heterogeneity and ratio of inner to outer radii on the buckling load is examined. Finite element and analytical approaches are discussed. It is shown that if the elasticity modulus decreases away from the center of a plate, the critical pressure for unsymmetric buckling is sufficiently lower than for a plate with constant mechanical properties. For solid plates—whether homogeneous or non-uniform and weakened toward the edge—the results obtained by both methods are close to each other. The calculations of the critical load for annular plates, conducted using the ANSYS finite element package and shell elements, show qualitative agreement with the analytical solutions of the problem.