We will 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 will obtain a new proof of existence of solutions for Liouville type mean-field equations in a supercritical regime. Moreover, we will also discuss an extension of this result to the case of singular Liouville equations.
NB:This talk is part of the activity of the MIUR Excellence Department Project MATH@TOV CUP E83C23000330006