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In this paper we formulate the equilibrium equation of a beam made ofgraphene material subjected to some boundary conditions and acted upon by axialcompression and nonlinear lateral constrains as a fourth-order nonlinearboundary value problem. We also formulate the nonlinear eigenvalue for bucklinganalysis of the beam. We verify the solvability of the buckling problem as anasymptotic expansion in a ratio of the elastoplastic parameters, that thespectrum is bounded away from zero and contains a discrete infinite sequence ofeigenvalues.We also verify, for certain ranges of the lateral forces, thesolvability of the general equations using energy methods and a suitableiteration scheme.

On the Solvability of an Euler Graphene Beam Subject to Axial Compressive Load