The necessary and sufficient condition for bifurcation in the von K�rm�n equations

Abstract

The paper is devoted to the study of bifurcation in the von Karman equations with two parameters $ \alpha, \beta\in \mathbb{R}_{+} $ that describe the behaviour of a thin round elastic plate lying on an elastic base under the action of a compressing force. The problem appears in the mechanics of elastic constructions. We prove the necessary and sufficient condition for bifurcation at points of the set of trivial solutions. Our proof is based on reducing the von Karman equations to an operator equation in Banach spaces with a nonlinear Fredholm map of index 0 and applying the Crandall-Rabinowitz theorem on simple bifurcation points or a finite-dimensional reduction and degree theory.