We are interested in several questions raised mainly by Mancini and
Martinazzi (2017) (see also work of McLeod and Peletier (1989) and
Pruss (1996)). We consider the perturbed Moser–Trudinger inequality
at the
critical level
,
where
,
satisfying
as
,
can be seen as a perturbation with respect to the original case
.
Under some additional assumptions, ensuring basically that
does not oscillate too
fast as
, we identify
a new condition on
for this inequality to have an extremal. This condition covers the case
studied by Carleson and Chang (1986), Struwe (1988), and Flucher (1992). We
prove also that this condition is sharp in the sense that, if it is not satisfied,
may
have no extremal.
Keywords
Moser–Trudinger inequality, blow-up analysis, elliptic
equations, extremal function
