As known, the constitutive relations of the Gurtin–Murdoch model of the surface elasticity take into account both elastic properties of a solid and surface tension of a liquid. Previously, most authors believed that, surface tension does not significantly affect the mechanical properties of a thin-walled elastic nano-objects and therefore it either was not or only in-plane taken into account, ignoring the Young–Laplace law in the transverse direction. In the present work, on the basis of the nonlinear representation of the surface tension obtained earlier, the structure of the strain energy of a nano-plate is substantiated, taking into account the surface tension in both tangential and transverse directions. Using the example of compressive buckling of a nano-plate, it is shown that the accounted surface tension terms can increase the size effect of the critical load up to 80%.