Multiple bifurcation in the solution set of the von Kármán equations with $S^{1}$-symmetries

Abstract

In this work we study bifurcation of forms of equilibrium of a thin circular elastic plate lying on an elastic base under the action of a compressive force. The forms of equilibrium may be found as solutions of the von Kármán equations with two real positive parameters defined on the unit disk in $\mathbb R^2$ centered at the origin. These equations are equivalent to an operator equation $F(x,p)=0$ in the real Hölder spaces with a nonlinear $S^{1}$-equivariant Fredholm map of index $0$. For the existence of bifurcation at a point $(0,p)$ it is necessary that $\dim\operatorname{Ker}F_{x}^{\prime}(0,p)>0$. The space $\operatorname{Ker}F_{x}^{\prime}(0,p)$ can be at most four-dimensional. We apply the Crandall-Rabinowitz theorem to prove that if $\dim\operatorname{Ker}F_{x}^{\prime}(0,p)=3$ then there is bifurcation of radial solutions at $(0,p)$. What is more, using the Lyapunov-Schmidt finite-dimensional reduction we investigate the number of branches of radial bifurcation at $(0,p)$.