New existence results for the mean field equation on compact surfaces via degree theory
Rendiconti del Seminario Matematico della Università di Padova, Tome 136 (2016), pp. 11-17.

We consider the following class of equations with exponential nonlinearities on a closed surface Σ:

-Δu=ρ 1 he u Σ he u dV g -1 |Σ|-ρ 2 he -u Σ he -u dV g -1 |Σ|,
which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here h is a smooth positive function and ρ 1 ,ρ 2 two positive parameters. By considering the parity of the Leray–Schauder degree associated to the problem, we prove solvability for ρ i (8πk,8π(k+1)),k. Our theorem provides a new existence result in the case when the underlying manifold is a sphere and gives a completely new proof for other known results.

DOI : 10.4171/RSMUP/136-2
Classification : 35
Mots clés : Geometric PDEs, Leray–Schauder degree, mean field equation
Jevnikar, Aleks 1

1 Università di Roma 'Tor Vergata', ROMA, ITALY
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     author = {Jevnikar, Aleks},
     title = {New existence results for the mean field equation on compact surfaces via degree theory},
     journal = {Rendiconti del Seminario Matematico della Universit\`a di Padova},
     pages = {11--17},
     publisher = {European Mathematical Society Publishing House},
     address = {Zuerich, Switzerland},
     volume = {136},
     year = {2016},
     doi = {10.4171/RSMUP/136-2},
     url = {http://www.numdam.org/articles/10.4171/RSMUP/136-2/}
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Jevnikar, Aleks. New existence results for the mean field equation on compact surfaces via degree theory. Rendiconti del Seminario Matematico della Università di Padova, Tome 136 (2016), pp. 11-17. doi : 10.4171/RSMUP/136-2. http://www.numdam.org/articles/10.4171/RSMUP/136-2/

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