Liposomes are small artificial vesicles of lipid bilayer, wich enclose and are surrounded by water. Morphological transformations in liposomes, starting from a spherical shape, due to changes in the osmotic pressure, have been described in the literature. The first transformation is into a circular biconcave form, afterwards the biconcave side view is maintained, while the front view reveals transformations into elliptical or regular polygonal forms, usually triangular, square or pentagonal. Finite elasticity and the theory of thin shells were used to analyse the behavior of the liposomes under decreasing volume. The biological membrane was considered as a two dimensional fluid layer, exhibiting solid properties to some extent, e.g., elasticity. The stability of the liposmes was studied by using the method of elastic perturbation to obtain the critical pressure for the biconcave transformation and the long liposome tubes. The transformations to elliptical and regular polygonal forms were studied using the linear stability equations of elasticity.