A mean field equation involving positively supported probability measures: blow-up phenomena and variational aspects

TL;DR

This paper investigates a mean field elliptic equation modeling vortex turbulence, analyzing blow-up phenomena and establishing solution existence via variational methods and improved inequalities.

Contribution

It introduces new insights into blow-up behavior and develops a variational approach for solutions in non-coercive regimes for a specialized mean field equation.

Findings

Description of blow-up phenomena and differences from standard equations

Existence of solutions in non-coercive regimes using improved inequalities

Analysis of Moser-Trudinger inequality related to blow-up masses

Abstract

We are concerned with an elliptic problem which describes a mean field equation of the equilibrium turbulence of vortices with variable intensities. In the first part of the paper we describe the blow-up phenomenon and highlight the differences from the standard mean field equation. In the second part we discuss the Moser-Trudinger inequality in terms of the blow-up masses and get the existence of solutions in a non-coercive regime by means of a variational argument, which is based on some improved Moser-Trudinger inequalities.