  In conformal geometry, the Compactness Conjecture asserts that the set of
Yamabe metrics on a smooth, compact, aspherical Riemannian manifold (M,g) is
compact. Established in the locally conformally flat case by Schoen [43,44] and
for n\leq 24 by Khuri-Marques-Schoen [26], it has revealed to be generally
false for n\geq 25 as shown by Brendle [8] and Brendle-Marques [9]. A stronger
version of it, the compactness under perturbations of the Yamabe equation, is
addressed here with respect to the linear geometric potential n-2/4(n-1)
Scal_g, Scal_g being the Scalar curvature of (M,g). We show that a-priori
L^\infty-bounds fail for linear perturbations on all manifolds with n\geq 4 as
well as a-priori gradient L^2--bounds fail for non-locally conformally flat
manifolds with n\geq 6 and for locally conformally flat manifolds with n\geq 7.
In several situations, the results are optimal. Our proof combines a finite
dimensional reduction and the construction of a suitable ansatz for the
solutions generated by a family of varying metrics in the conformal class of g.
