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

Jevnikar, Aleks

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 $\Sigma$:

$$-\Delta u={\rho }_1\left(\frac{h{e}^u}{{\int }_{\Sigma }h{e}^{u}d{V}_g}-\frac{1}{\left|\Sigma \right|}\right)-{\rho }_2\left(\frac{h{e}^{-u}}{{\int }_{\Sigma }h{e}^{-u}d{V}_g}-\frac{1}{\left|\Sigma \right|}\right),$$

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.