Full Palais–Smale compactness for the critical elliptic system

Prove the full Palais–Smale condition for the critical cooperative and non-cooperative elliptic system with general subcritical perturbation, beyond the established result that every Palais–Smale sequence at a level below \(\frac{1}{N}S^{N/2}\) has a nontrivial weak limit that is a critical point.

Background

The paper addresses the loss of compactness caused by the critical Sobolev exponent in a two-component elliptic system whose linear coupling is represented by a symmetric matrix and whose lower-order perturbation may be general and sign-changing. Instead of proving global Palais–Smale compactness, the authors establish a restricted nonvanishing result below the critical energy level $\frac{1}{N}S^{N/2}$. The unresolved issue explicitly identified by the authors is whether the full Palais–Smale condition can be proved for the associated energy functional, including levels at or above this critical threshold.