### Minimizers and symmetric minimizers for problems with critical Sobolev exponent

DOI: http://dx.doi.org/10.12775/TMNA.2009.044

#### 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.

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.

#### Keywords

Concentration-compactness principle; critical Sobolev exponent; symmetric solutions of elliptic equations; Sobolev embeddings in weighted spaces

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