**Existence of maximizers for scaling invariant ***L*^{p}
inequalities

We summarize methods to prove that best constants are attained
for certain inequalities which are most useful in PDEs.
The earliest systematic results have been obtained by Lieb in 1983.
They apply e.g. to the Hardy-Littlewood-Sobolev inequality
and related estimates (like the classical Sobolev inequality).
Later a further approach has been developed by P.-L. Lions
which allowed to deal with inequalities having no
favourable properties under rearrangement.
In a final section we present a recent new method
to show that the best constant is attained in the standard
Strichartz estimate for the Schrödinger equation.

*Markus Kunze*

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