Minimizers and symmetric minimizers for problems with critical Sobolev exponent

Shoyeb Waliullah

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.

Keywords


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

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