Compactness and quasilinear problems with critical exponents

Abstract

A compactness result is revised in order to prove the pointwise convergence of the gradients of a sequence of solutions to a general quasilinear inequality (anisotropic or not, degenerate or not) and for an arbitrary open set. Combining this result with the well-known Brézis-Lieb lemma, we derive simple proofs of Palais-Smale properties in many optimization problems especially on unbounded domains.