Inserted: 28 may 2016
Last Updated: 9 jan 2017
Journal: Proc. Royal Soc. Edinburgh Sect. A
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