The classical nonlinear von Kármán theory is adapted for modeling the behavior of nanoplates with surface stresses taken into account according to the strain-consistent Gurtin-Murdoch model of the surface elasticity using the effective elastic moduli. This allows us to apply the known solutions and methods for macroplates to nanoplates. In particular, the problem of buckling an thin elastic annular plate stretched by two point loads has been solved by the authors early. In this paper, the same problem are solving for an annular plate having nanosize. In contrast to the previous studies, where the size effect under buckling of nanoplates was considered mainly in the case of a homogeneous stress field, here, the size effect is considered when a nanoplate is buckling under an inhomogeneous stress field with a singularity.