On the existence of solutions to the generalized Marguerre-von Karman equations

Using techniques from asymptotic analysis, the second author has recently identified equations that generalize the classical Marguerre-von Karman equations for a nonlinearly elastic shallow shell by allowing more realistic boundary conditions, which may change their type along the lateral face of the shell. We first reduce these more general equations to a single "cubic" operator equation, whose sole unknown is the vertical displacement of the shell. This equation generalizes a cubic operator equation introduced by A S. Berger and P Fife for analyzing the von Karman equations for a nonlinearly elastic plate. We then establish the existence of a solution to this operator equation by means of a compactness method due to J. L. Lions.