A positive solution for some critical p-Laplacian systems

This paper deals with the existence of a positive solution for two classes of critical quasilinear system. We prove these results by a variant of mountain pass lemma, combining two convergence theorems and two estimate results. Here we avoid the usual compactness arguments (e.g., Palais-Smale condition or Cerami condition) and reveal the potential of some energy level estimates for the existence of nontrivial solutions.