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

Analytic aspects of the Tzitzéica equation: blow-up analysis and existence results We are concerned with the following class of equations with exponential nonlinearities: $$\begin{aligned} \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mathrm {in}~B_1\subset \mathbb {R}^2, \end{aligned}$$ Δ u + h 1 e u - h 2 e - 2 u = 0 in B 1 ⊂ R 2 , which is related to the Tzitzéica equation. Here $$h_1, h_2$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

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

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References (69)

Publisher
Springer Journals
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
DOI
10.1007/s00526-017-1136-6
Publisher site
See Article on Publisher Site

Abstract

We are concerned with the following class of equations with exponential nonlinearities: $$\begin{aligned} \Delta u+h_1e^u-h_2e^{-2u}=0 \qquad \mathrm {in}~B_1\subset \mathbb {R}^2, \end{aligned}$$ Δ u + h 1 e u - h 2 e - 2 u = 0 in B 1 ⊂ R 2 , which is related to the Tzitzéica equation. Here $$h_1, h_2$$ 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.

Journal

Calculus of Variations and Partial Differential Equations – Springer Journals

Published: Mar 17, 2017

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