Symmetry, quantitative Liouville theorems and analysis of large solutions of conformally invariant fully nonlinear elliptic equations

TL;DR

This paper analyzes the blow-up behavior of solutions to conformally invariant fully nonlinear elliptic equations, establishing profiles, bounds, and a quantitative Liouville theorem to understand large solutions and their asymptotics.

Contribution

It provides a comprehensive analysis of blow-up profiles and distances for solutions, introducing universal bounds and a quantitative Liouville theorem for these equations.

Findings

Bounded the distance between blow-up points by a universal constant

Showed solutions approximate a standard bubble near blow-up points

Established comparability of bubble heights via a universal factor

Abstract

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points is bounded from below by a universal positive number, (ii) the solutions are very close to a single standard bubble in a universal positive distance around each blow-up point, and (iii) the heights of these bubbles are comparable by a universal factor. As an application of this result, we establish a quantitative Liouville theorem.