An Existence Result for the Mean Field Equation on Compact Surfaces in a Doubly Supercritical Regime

We consider a class of variational equations with exponential nonlinearities on a compact Riemannian surface, describing the mean field equation of the equilibrium turbulance with arbitrarily signed vortices. For the first time, we consider the problem with both supercritical parameters and we give an existence result by using variational methods. In doing this, we present a new Moser-Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of both e^u and e^{-u}, where u is the unknown function in the equation.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1105.3701 by other authors. The proof of Lemma 3.9 has been fixe

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

We consider a class of equations with exponential non-linearities on a compact surface which arises as the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We prove an existence result via degree theory. This yields new existence results in case of a topological sphere. The proof is carried out by considering the parity of the Leray-Schauder degree associated to the problem. With this method we recover also some known previous results

Analytic aspects of the Tzitz\'eica equation: blow-up analysis and existence results

We are concerned with the following class of equations with exponential nonlinearities: \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, which is related to the Tzitz\'eica equation. Here $h_1, h_2$ are two smooth positive functions. The purpose of the paper is to initiate the analytical study of the above equation and to give a quite complete picture both for what concerns the blow-up phenomena and the existence issue. In the first part of the paper we provide a quantization of local blow-up masses associated to a blowing-up sequence of solutions. Next we exclude the presence of blow-up points on the boundary under the Dirichlet boundary conditions. In the second part of the paper we consider the Tzitz\'eica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result

A note on a multiplicity result for the mean field equation on compact surfaces

We are concerned with a Liouville-type equation with exponential nonlinearities on a compact surface which describes the mean field equation of the equilibrium turbulence with arbitrarily signed vortices. We provide the first multiplicity result for this class of equations by using Morse theory

A topological join construction and the Toda system on compact surfaces of arbitrary genus

We consider a Toda system of Liouville equations defined on a compact surface which arises as a model for non-abelian Chern-Simons vortices. For the first time the range of parameters $\rho_1 \in (4k\pi , 4(k+1)\pi)$, $k \in \mathbb{N}$, $\rho_2 \in (4\pi, 8\pi )$ is studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by means of a new improved Moser-Trudinger type inequality and introducing a topological join construction in order to describe the interaction of the two components