One of the main problems encountered in solving variational problems is to show the convergence of minimizing sequence (the direct method). In many problems, it is relatively easy to show the boundedness of a minimizing sequence in an appropriate Sobolev space.
Th aim of this section is to derive the concentration-compactness at infinity (just for subcritical case). Roughly speaking, for unbounded domains due to the lack of compact embedding in the Sobolev embedding theorem, the loss of mass of a weakly convergent sequence may occur. In fact, this can only occur at infinity.
Let us firstly consider the case is a bounded domain. Let be a sequence in such that
weakly in (i.e. and for any ). Assume . The Sobolev embedding tells us that
is compact. Thus
strongly in in the sense that
However, when is unbounded, the situation is quite different. Actually, we are going to prove the following
Theorem. Let be an unbounded domain and . Let be a sequence in such that in and define
Then these quantities are well-defined and satisfy
Proof. For each
Letting gives us the first relation.
in . Using the weak lower semicontinuity of norms we get
Thus by sending we obtain the second identity.