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

TL;DR

This paper analyzes the blow-up behavior and existence of solutions for a class of exponential nonlinear equations related to the Tzitzéica equation, providing quantization results, boundary behavior analysis, and existence theorems.

Contribution

It offers a comprehensive analytical framework for the Tzitzéica equation, including blow-up quantization, boundary behavior exclusion, and existence results on compact surfaces.

Findings

Quantization of local blow-up masses

Exclusion of boundary blow-up points under Dirichlet conditions

Existence results for solutions on compact surfaces

Abstract

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.