In this discussion, we will prove the following theorem.

**Theorem (Sharp Sobolev Embedding, Aubin & Talenti)** * Fix . The inequality *

*
** where the constant is determined by inserting . Moreover, equality occurs in the above if and only if for some , , and . *

The most difficult part of the proof is to show that there is indeed an optimizer (cf previous discussion with the sharp Gagliardo–Nirenberg inequality). Moreover, once we have the correct “bubble decomposition’, the existence of an optimizer is then a simple sub-additivity argument.

**Theorem ** * Let be a bounded sequence in with . Then there exist , , , so that along a subsequence in we can write *

*
* for each finite , with the following properties

- in .
- .
- For all , .
- .
- .

* *

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