On the bifurcation of solutions to Marguerre–von Kármán equations

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The objective of this work is to study the bifurcation of solutions of the Marguerre–von Kármán equations, which constitute a mathematical model for the buckling of Marguerre–von Kármán shallow shells. More precisely, we reduce the Marguerre–von Kármán equations to a single equation with a cubic operator; its second member depends on the function that defines the middle surface of the shallow shell and the applied forces. Next, we prove a general existence theorem for the reduced equation, by using the main theorem on pseudomonotone operators. Then we study the bifurcation of solutions in the reduced equation, with a second member, is small, at neighborhood of the simple characteristic value of the linearized problem.