We introduce a notion of cocompact imbeddings relative to a group of linear isometries.We discuss the notion of critical Sobolev nonlinearity in connection with the usual dilation actions that make the (non-compact) limit Sobolev imbedding co-compact and yield solutions of Talenti type for semilinear elliptic equations with self-similar autonomous nonlinearities of critical growth.We then consider similar dilation and translations groups for $H_01(B)$, where $B$ is a unit disk on a plane, which preserve the Sobolev norm, but do not preserve the Trudinger-Moser functional $\int e^{4\pi u^2}$. We give then two examples of invariant critical nonlineairites that are stronger than Trudinger-Moser nonlinearity and lack the weakly continuity properties of the latter.
We give further examples of cocompactness in Sobolev spaces over manifolds, including subelliptic spaces over nilpotent Lie groups, as well as some interpolation results that lead to cocompactness of imbeddings of Besov spaces. This work is partially done in collaboration with Adimurthi, M. Cwikel and J.M. do O.