Non-compactness and multiplicity results for the Yamabe problem on Sn

By Massimiliano Berti, A. Malchiodi. 2001

48 citations recorded by the source corpus.

The aim of this paper is to show the existence of metrics ge on Sn, where ge is a perturbation of the standard metric g0, for which the Yamabe problem possesses a sequence of solutions unbounded in L∞(Sn). The metrics ge that we find are of class Ck on Sn with (k⩽n−34). We also prove some new multiplicity results.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/dc3d86e963ba12e0a0804b93e3487dab8b998abf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Bifurcation and local rigidity of homogeneous solutions to the Yamabe problem on spheres (2011, background): “Later, Berti and Malchiodi [6] obtained the existence of a sequence of unbounded solutions (in the L∞ sense) among other multiplicity results, also for perturbations of the round metric.”
Blow-up phenomena for the Yamabe equation (2007, background): “Given positive integers n and k such that k ≥ 2 and n ≥ 4k + 3, Berti and Malchiodi showed that there exists a Riemannian metric g on Sn (of class Ck) for which the set of solutions to the Yamabe PDE (1) fails to be compact (see [6], Theorem 1.”
Blow-up phenomena for the Yamabe equation II (2009, unknown): “This result, combined with the positive mass theorem, implies the Compactness Conjecture for those dimensions.”
Fractional elliptic problems with critical growth in the whole of $\R^n$ (2015, background): “In the classical case, that is when s = 1 and the fractional Laplacian boils down to the classical Laplacian, there is an intense literature regarding this type of problems, see [1, 2, 3, 4, 5, 6, 9, 13, 18, 22, 24, 38, 39], and references therein.”