Compact embeddings of Sobolev spaces with power weights perturbed by slowly varying functions

Abstract

We study compact embeddings of weighted Sobolev spaces into Lebesgue spaces on the unit ball in . The weight is of slowly varyingly disturbed polynomial growth with a singularity at the origin. It extends [21], [27] to a wider class of weights. Special attention is paid to the influence of the growth rate of the weight on the quality of compactness, measured in terms of entropy and approximation numbers. In case of Hilbert spaces, the results are related to the distribution of eigenvalues of some degenerate elliptic operators.