In conformal geometry, the Compactness Conjecture asserts that the set of Yamabe metrics on a smooth, compact Riemannian manifold (M,g) is compact. Established in the locally conformally flat case [41,42] and for n\leq 24 [23], it has revealed to be generally false for n\geq 25 [8,9]. 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). Even tough the Yamabe equation is compact in some cases, surprisingly we show that a-priori L^\infty-bounds fail on all manifolds with n\geq 4 as well as H_1^2-bounds do in the locally conformally flat case when n\geq 7. In several situations, the results are optimal

Given (M, g) a smooth, compact Riemannian n-manifold, we consider equations like ∆ g u + hu = u 2 * −1−ε , where h is a C 1-function on M , the exponent 2 * = 2n/ (n − 2) is critical from the Sobolev viewpoint, and ε is a small real parameter such that ε → 0. We prove the existence of blowing-up families of positive solutions in the subcritical and supercritical case when the graph of h is distinct at some point from the graph of n−2 4(n−1) Scal g .

We establish pointwise a priori estimates for solutions in D 1,p (R n ) of equations of type −∆ p u = f (x, u), where p ∈ (1, n), ∆ p := div |∇u| p−2 ∇u is the p-Laplace operator, and f is a Caratheodory function with critical Sobolev growth. In the case of positive solutions, our estimates allow us to extend previous radial symmetry results. In particular, by combining our results and a result of Damascelli-Ramaswamy [5], we are able to extend a recent result of Damascelli-Merchán-Montoro-Sciunzi [6] on the symmetry of positive solutions in D 1,p

We prove bounded stability for strongly coupled critical elliptic systems in the inhomogeneous context of a compact Riemannian manifold when the potential of the operator is less, in the sense of bilinear forms, than the geometric threshold potential of the conformal Laplacian.

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