Minimizers and symmetric minimizers for problems with critical Sobolev exponent

In this paper we will be concerned with the existence and non-existence of constrained minimizers in Sobolev spaces D(R ), where the constraint involves the critical Sobolev exponent. Minimizing sequences are not, in general, relatively compact for the embedding D(R) →֒ L ∗ (R , 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(R ) exists so that all minimizing sequences are relatively compact. In fact we give an example of a Q and an equivalent norm in D(R ) so that all minimizing sequences are relatively compact.