JournalsrsmupVol. 136pp. 11–17

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

  • Aleks Jevnikar

    Università di Roma 'Tor Vergata', Italy
New existence results for the mean field equation on compact surfaces via degree theory cover
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Abstract

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

Δu=ρ1(heuΣheudVg1Σ)ρ2(heuΣheudVg1Σ),- \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 hh is a smooth positive function and ρ1,ρ2\rho_1, \rho_2 two positive parameters. By considering the parity of the Leray–Schauder degree associated to the problem, we prove solvability for ρi(8πk,8π(k+1)),kN\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.

Cite this article

Aleks Jevnikar, New existence results for the mean field equation on compact surfaces via degree theory. Rend. Sem. Mat. Univ. Padova 136 (2016), pp. 11–17

DOI 10.4171/RSMUP/136-2