We have studied, by grazing incidence x-ray diffraction, the surface stress variation under nitrogen and oxygen adsorption on Cu(001), exploiting the nitrogen self-organization on Cu(001). The measured structure factors are analyzed in the framework of linear elasticity theory. This allows us to calculate the elastic displacements in the bulk from the surface stress discontinuities located at the domain boundaries between the different phases that are present at the surface. We find that the variations of surface stress upon adsorption are of the order of 5.5 Nm(-1) for the N/Cu(001)-c(2x2) structure and 1.2 Nm(-1) for O/Cu(001)-c(2x2) structure. For both cases, elastic interactions due to surface stress variations are higher than electrostatic interactions. In addition, we have precisely determined the atomic structure of the c(2x2) N/Cu(001) and O/Cu(001) (2 root 2 x root 2)R45 degrees structures, showing that there is no significant Cu buckling in the c(2x2) N/Cu(001) case.