Rendiconti del Seminario Matematico della Università di Padova


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Volume 136, 2016, pp. 11–17
DOI: 10.4171/RSMUP/136-2

New existence results for the mean field equation on compact surfaces via degree theory

Aleks Jevnikar[1]

(1) Università di Roma 'Tor Vergata', Italy

We consider the following class of equations with exponential nonlinearities on a closed surface $\Sigma$: $$ - \Delta u = \rho_1 \left( \frac{h \,e^{u}}{\int_\Sigma h \,e^{u} \,dV_g} - \frac{1}{|\Sigma|} \right) - \rho_2 \left( \frac{h \,e^{-u}}{\int_\Sigma h \,e^{-u} \,dV_g} - \frac{1}{|\Sigma|} \right),$$ which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. Here $h$ is a smooth positive function and $\rho_1, \rho_2$ two positive parameters. By considering the parity of the Leray–Schauder degree associated to the problem, we prove solvability for $\rho_i \in (8\pi k, 8\pi(k+1)),\, k \in \mathbb N$. 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.

Keywords: Geometric PDEs, Leray–Schauder degree, mean field equation

Jevnikar Aleks: New existence results for the mean field equation on compact surfaces via degree theory. Rend. Sem. Mat. Univ. Padova 136 (2016), 11-17. doi: 10.4171/RSMUP/136-2