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

作者：Jevnikar Aleks^*; Yang Wen

来源：Calculus of Variations and Partial Differential Equations, 2017, 56(2): 43.

DOI：10.1007/s00526-017-1136-6

We are concerned with the following class of equations with exponential nonlinearities:
Delta u + h(1)e(u) - h(2)e(-2u) = 0 in B1 subset of R-2,
which is related to the Tzitzeica 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 Tzitzeica equation on compact surfaces: we start by proving a sharp Moser-Trudinger inequality related to this problem. Finally, we give a general existence result.

• 出版日期2017-4