*Accepted Paper*

**Inserted:** 6 may 2016

**Last Updated:** 9 jan 2017

**Journal:** Calc. Var. and PDEs

**Year:** 2016

**Abstract:**

We are concerned with the following class of equations with exponential nonlinearities: $\Delta u+h_1e^u-h_2e^{-2u}=0$ in $B_1 \subset\mathbb{R}^2,$ which is related to the Tzitzéica 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 Tzitzéica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result.

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