In this entry, we continue to talk about the Concentration-Compactness Principle discovered by P.L. Lions [here]. In the previous entry, we already discussed two forms of non-compactness due to unbounded domains. Here we discuss what happens when passing to the limit on those functionals along weakly convergent subsequences.
Theorem (Lions). Let be sequence in weakly convergent to and such that
- converges weak* to a nonnegative measure ,
- converges weak* to a nonnegative measure .
Then there exists an at most countable index set , sequence , , , , such that
where is the best Sobolev constant and are Dirac measures assigned to . If and
then is a singleton and
for some .
Apparently, the theorem does not provide any information about possible loss of mass at infinity of a weakly convergent minimizing sequence. We shall consider that case in the forthcoming topic.
- Concentration-Compactness principle, I.
- Concentration-Compactness Principle: The loss of mass at infinity in the subcritical case.
- Jan Chabrowski, Variational methods for potential operator equations, Walter de Gruyter, 1997.