Abstract: In this talk, we present some results on the critical points to the general- ized Ginzburg-Landau equations in dimensions n≥ 3 which satisfy a suitable energy bound, but are not necessarily energy-minimizers. When the parameter in the equa- tions tend to zero, such solutions are shown to converge to singular n-harmonic maps into spheres which are conformally invariant, and the convergence is strong away from a finite set consisting 1) of the infinite energy singularities of the limiting map, and 2) of points where bubbling off of finite energy n-harmonic maps takes place. The latter case is specific to dimensions greater than 2.