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The necessary and sufficient condition for bifurcation in the von Kármán equations. (English) Zbl 1092.35112
Summary: The paper is devoted to the study of bifurcation in the von Kármán 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 Kármán 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.

MSC:
 35Q72 Other PDE from mechanics (MSC2000) 46T99 Nonlinear functional analysis
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