Minimizers and symmetric minimizers for problems with critical Sobolev exponent

Abstract

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