Multiple bifurcation in the von Karman equations

We numerically investigate nontrivial solutions for some thin plate problems. We find that at a corank-2 bifurcation point of the von Karman equations with simply supported boundary conditions, four nontrivial solution branches bifurcate. Thus the results of Holder, Schaeffer, and Golubitsky are numerically verified, where two symmetry-breaking solution branches are obtained. We also show that the result of Allgower, Bohmer, and Mei in second-order semilinear elliptic problems can be extended to certain fourth-order elastic plate problems. The local bifurcation behaviors of the former and the latter are different. The practical implementations are given. Sample numerical results are reported.