Compactness of Palais-Smale sequences with controlled Morse Index for a Liouville type functional

TL;DR

This paper demonstrates that Palais-Smale sequences with bounded Morse index are precompact for Liouville type functionals on closed surfaces, leading to new existence proofs for solutions in supercritical regimes and extensions to singular cases.

Contribution

It establishes precompactness of Palais-Smale sequences under Morse index bounds and provides new existence proofs for Liouville mean-field equations, including singular cases.

Findings

Palais-Smale sequences with bounded Morse index are precompact.

New proof of existence for supercritical Liouville mean-field equations.

Extension of results to singular Liouville equations.

Abstract

We prove that Palais-Smale sequences for Liouville type functionals on closed surfaces are precompact whenever they satisfy a bound on their Morse index. As a byproduct, we obtain a new proof of existence of solutions for Liouville type mean-field equations in a supercritical regime. Moreover, we also discuss an extension of this result to the case of singular Liouville equations.