A Variational Analysis of the Toda System on Compact Surfaces. Issue 3 (14th November 2012)

• Record Type: Journal Article
• Authors: Malchiodi, Andrea; Ruiz, David
• Is Part Of: Communications on pure and applied mathematics. Volume 66:Issue 3(2013:Mar.)
• Page Start: 332
• Page End: 371
• Publication Date: 2012-11-14
• Subjects: Mathematics -- Periodicals; Mechanics -- Periodicals; Mathématiques -- Périodiques; Mécanique -- Périodiques; 510.5
• DOI: 10.1002/cpa.21433
• Languages: English
• ISSNs: 0010-3640

Abstract

<abstract abstract-type="main" xml:lang="en"> <title>Abstract</title> <p>In this paper we consider the following <italic>Toda system</italic> of equations on a compact surface: <disp-formula content-type="mathematics" id="di-ueqn-1"><alternatives><graphic position="anchor" mimetype="image" xlink:href="ark:/27927/pgg1n04h8tz" orientation="portrait" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><tex-math notation="LaTeX"><![CDATA[\documentclass{article}\usepackage{mathrsfs}\usepackage{amsmath}\pagestyle{empty}\begin{document}\begin{align*} \begin{cases} - \Delta u_1 = 2 \rho_1 \Bigl( \frac{h_1 e^{u_1}}{\int \sum h_1 e^{u_1} dV_g} - 1 \Bigr) - \rho_2 \Bigl( \frac{h_2 e^{u_2}}{\int\sum h_2 e^{u_2} dV_g} - 1 \Bigr), \\[2\jot] - \Delta u_2 = 2 \rho_2 \Bigl( \frac{h_2 e^{u_2}}{\int\sum h_2 e^{u_2} dV_g} - 1 \Bigr) - \rho_1 \Bigl( \frac{h_1 e^{u_1}}{\int\sum h_1 e^{u_1} dV_g} - 1 \Bigr) \end{cases}. \end{align*} \end{document}]]></tex-math></alternatives></disp-formula> We will give existence results by using variational methods in a noncoercive case. A key tool in our analysis is a new Moser‐Trudinger type inequality under suitable conditions on the center of mass and the scale of concentration of the two components <italic>u</italic>₁ and <italic>u</italic>₂. © 2012 Wiley Periodicals, Inc.</p> </abstract>