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Application of topological degree to the study of bifurcation in von Kármán equations. (English) Zbl 1006.35093
The author studies a bifurcation problem in von Kármán equations for thin elastic circular plate lying on the elastic base under the action of a compressing force. The problem is described by the equations $\Delta \Delta w-[w,\sigma] +2\alpha\Delta w+\beta w-\gamma w^3=0 \text{ in } \Omega,\tag{1}$ $\Delta\Delta \sigma+0.5[w,w]=0\text{ in }\Omega,\tag{2}$ where $$\Omega$$ is the unit ball in $$\mathbb{R}^2$$, $$\Delta$$ the Laplace operator, $$w$$ the deflection, $$\sigma$$ the stress function, $$\alpha$$ the applied compressing force, $$\beta,\gamma$$ the parameters of elastic foundation, and $[h,g]= {\partial^2h \over\partial x^2} {\partial^2g\over \partial y^2}-2{\partial^2 h\over \partial x\partial y}{\partial^2 g\over\partial x\partial y}+{\partial^2 h\over\partial y^2}{\partial^2g\over\partial x^2}.$ The von Kármán equations (1), (2) are reformulated into an operator equation $$F(x,p)=0$$ in a real Banach space, $$x=(w,\sigma)$$, $$p=(\alpha,\beta)$$. It is shown that the necessary condition for bifurcation at $$(0,p)$$ is that $$N(p)=\dim(\text{Ker} F_x'(0,p))>0$$. The author proves $$N(p)<4$$. To solve the bifurcation problem homotopy and degree theory are used.

##### MSC:
 35Q72 Other PDE from mechanics (MSC2000) 74K20 Plates 74G60 Bifurcation and buckling
##### Keywords:
bifurcation; von Kármán equations; topological degree
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