Calculus of Variations and Geometric Measure Theory

A. Jevnikar - W. Yang

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

created by jevnikar on 28 May 2016
modified on 09 Jan 2017


Accepted Paper

Inserted: 28 may 2016
Last Updated: 9 jan 2017

Journal: Proc. Royal Soc. Edinburgh Sect. A
Year: 2016


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.

Keywords: Geometric PDEs, Mean field equation, Blow-up analysis, Variational methods