Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results

Abstract

We are concerned with the following class of equations with exponential nonlinearities: $$ Δu+h_1e^u-h_2e^{-2u}=0 \qquad \mbox{in } B_1\subset\mathbb{R}^2, $$ 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.

References in corpus (3)

• Strominger--Yau--Zaslow geometry, Affine Spheres and Painlevé III
• A topological join construction and the Toda system on compact surfaces of arbitrary genus
• Concentrating solutions for a Liouville type equation with variable intensities in 2D-turbulence

Cited by in corpus (6)

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• Uniform bounds for higher-order semilinear problems in conformal dimension
• Wave equations associated to Liouville-type problems: global existence in time and blow up criteria