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

Part of: Miscellaneous topics - Partial differential equations Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  27 December 2018

Aleks Jevnikar  and
Wen Yang
Aleks Jevnikar
Affiliation:
Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa, Italy (aleks.jevnikar@dm.unipi.it)
Wen Yang
Affiliation:
Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, P. R. China (wyang@wipm.ac.cn)
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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 picture and highlight the differences from the standard mean field equation as we observe non-quantization phenomenon. 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 PDEsmean field equationblow-up analysisvariational methods

MSC classification

Primary: 35J61: Semilinear elliptic equations
Secondary: 35J20: Variational methods for second-order elliptic equations 35R01: Partial differential equations on manifolds 35B44: Blow-up
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 2 , April 2019 , pp. 325 - 352
Copyright
Copyright © Royal Society of Edinburgh 2018 

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