In this talk, the motivation and foundations of the alpha-energy functional, an approximation to the Dirichlet energy, will be presented. This alpha-energy satisfies the Palais-Smale condition, a kind of compactness condition. This guarantees the existence of a sequence of critical points with another critical point as a limit. However, these are not minima as e.g. in the theory of minimal surfaces, but saddle points. In the second part of the talk we consider a general sequence of critical points and study how the energy or the alpha approximation of the energy behaves under this convergence process. In doing so, we will come across the so-called bubbling phenomena according to J. Sacks and K. Uhlenbeck