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We investigate bifurcation in the solution set of the von K arm an equations on a disk R 2 with two positive parameters and . The equations describe the behaviour of an elastic thin round plate lying on an elastic base under the action of a compressing force. The method of analysis is based on reducing the problem to an operator equation in real Banach spaces with a nonlinear Fredholm map F of index zero (to be dened later) that depends on the parameters and . Applying the implicit function theorem we obtain the following necessary condition for bifurcation: if (0;p) is a bifurcation point then dim KerF 0 x(0;p) > 0. Next, we give a full description of the kernel of the Fr echet derivative of F. We study in detail the situation when the dimension of the kernel is one. We prove that (0;p) is a bifurcation point by the use of the Lyapunov{Schmidt nite-dimensional reduction and the Crandall{Rabinowitz theorem. For a one-dimensional bifurcation point, analysing the Lyapunov{Schmidt branching equation we determine the number of families of solutions, their directions and asymptotic behaviour (shapes).

Bifurcation in the solution set of the von Kármán equations of an elastic disk lying on an elastic foundation