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Compactness of sign-changing solutions to scalar curvature-type equations with bounded negative part.

Banff International Research Station for Mathematical Innovation and Discovery

We consider the equation $\triangle_g u+hu=|u|^{2^*-2}u$ in a closed Riemannian manifold $(M,g)$, where $h$ is some Holder continuous function in $M$ and $2^* = \frac{2n}{n-2}$, $n:=dim M$. We obtain a sharp compactness result on the sets of sign-changing solutions whose negative part is a priori bounded. We obtain this result under the conditions that $n \ge 7$ and $h<(n-2)/(4(n-1)) S_g$ in $M$, where $S_g$ is the Scalar curvature of the manifold. We show that these conditions are optimal by constructing examples of blowing-up solutions, with arbitrarily large energy, in the case of the round sphere with a constant potential function $h$. This is a joint work with J. V\'etois (McGill University, Montr\'eal)

Mathematics; Partial differential equations; Differential geometry

video/mp4

Author affiliation: Université Libre de Bruxelles

2019-11-04

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/72179

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/