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

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.