Theorem (Lions). Suppose that is a bounded sequence in , and let
for all . Then there exists a subsequence satisfying one of the three following possibilities
- (Compactness) There exists a sequence in such that is tight, that is, for every there exists such that
- (Dichotomy) There exists such that for all , there exist , bounded sequences and in satisfying for
with as and
We now briefly explain how the principle of loss of mass at infinity [here] leads to the above Theorem. Let be a bounded sequence in such that
for all . We may assume that
in . According to the principle of loss of mass at infinity, we have
We now distinguish three cases
- and thus (almost everywhere)
- and thus
- If this is the case, then either
for all or
for some . In the first case, by the Lions lemma [here], we deduce that
in for all . In the second case, there exists a subsequence of relabelled again by , and a sequence such that
- If this is the case, then the sequence is tight.
- If this is the case, we get the dichotomy property. The way to see this is to use the Lévy concentration functions defined by
Each function is increasing on and by the Helly selection theorem
is an increasing function on . Since , it is easy to check that
Remark. The dichotomy property tells us that sequence can be split into two parts concentrating on two disjoint sets whose distance tends to as .
Remark. It is easy to find that if compactness happens then neither vanishing nor dichotomy can happen, but vanishing and dichotomy can happen together. In fact, it had been proved that up to a subsequence of either compactness or dichotomy happens. This result was due to Xing, Yi and Yaotian [here].
In practice, we look for a minimizing sequence of an energy functional. We then try to identify which of compactness, vanishing or dichotomy happens. Once this step is done, we can solve the problem (most of the time, we wish the dichotomy cannot occur). I will demonstrate several applications later.
- 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.