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: $$\mathrm{\Delta }u+h_1{e}^u-h_2{e}^{-2u}=0in{B}_1\subset {\mathbb{R}}^2,$$ 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.