We consider the fractional critical problem
$A_{s}u=K(x)u^{(n+2s)/(n-2s)},u>0$
in
$\unicode[STIX]{x1D6FA},u=0$
on
$\unicode[STIX]{x2202}\unicode[STIX]{x1D6FA}$
, where
$A_{s},s\in (0,1)$
, is the fractional Laplace operator and
$K$
is a given function on a bounded domain
$\unicode[STIX]{x1D6FA}$
of
$\mathbb{R}^{n},n\geq 2$
. This is based on A. Bahri’s theory of critical points at infinity in Bahri [Critical Points at Infinity in Some Variational Problems, Pitman Research Notes in Mathematics Series, 182 (Longman Scientific & Technical, Harlow, 1989)]. We prove Bahri’s estimates in the fractional setting and we provide existence theorems for the problem when
$K$
is close to 1.