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

.

It now follows from the Brezis-Lieb lemma or the use of the Minkowski inequality

and

that

or

.

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

and

.

Then these quantities are well-defined and satisfy

and

.

*Proof*. For each

Letting gives us the first relation.

Similarly,

.

Note that

in . Using the weak lower semicontinuity of norms we get

which implies

.

Thus by sending we obtain the second identity.

### Like this:

Like Loading...

*Related*

## Leave a Reply