MINIMIZERS AND SYMMETRIC MINIMIZERS FOR PROBLEMS WITH CRITICAL SOBOLEV EXPONENT

In this paper we will be concerned with the existence and nonexistence of constrained minimizers in Sobolev spaces D-k,D-p (R-N), where the constraint involves the critical Sobolev exponent. Minimizing sequences are not, in general, relatively compact for the embedding D-k,D-p(R-N) hooked right arrow L-p*(R-N, Q) when Q is a non-negative, continuous, bounded function. However if Q has certain symmetry properties then all minimizing sequences are relatively compact in the Sobolev space of appropriately symmetric functions. For Q which does not have the required symmetry, we give a condition under which an equivalent norm in D-k,D-p(R-N) exists so that all minimizing sequences are relatively compact. In fact we give an example of a Q and an equivalent norm in D-k,D-p(R-N) so that all minimizing sequences are relatively compact.