Analytic aspects of the Toda system: I. A Moser‐Trudinger inequality

Abstract

In this paper, we analyze solutions of the open Toda system and establish an optimal Moser-Trudinger type inequality for this system.

Analytic aspects of the Toda system: II. Bubbling behavior and existence of solutions

“…Let Σ be a Riemann surface with Gaussian curvature K. We consider the following system N where (a ij ) is the inverse matrix of A. In [11], we proved that J ρ has a lower bound in H if and only if ρ i ≤ 4π for any i, which is the Moser-Trudinger inequality for the Toda system. From this inequality, we know that if ρ i < 4π (∀j) then J ρ iscoercive condition and hence J ρ has a minimizer, which certainly satisfies system (1.3).…”

“…However, the analysis of the Toda system becomes more difficult, because the basic analytic tool, the maximum principle, does not work. In [11], we established a Moser-Trudinger type inequality for the Toda system, while a rough bubbling behavior of solutions to system (1.1) was considered. See also [13].…”

“…The proof of Theorem 1.2 follows from the study of the convergence of the sequence of solutions, which was initiated in [11] for the Toda system. Let x be an blow-up point of the sequence u k , i.e., there exists a sequence x k → x such that max{u We call (σ 1 , σ 2 ) a blow-up value at x.…”

Asymptotic behavior of SU(3) Toda system in a bounded domain

“…Jost-Wang [12] and Jost-Lin-Wang [10] studied the SU(3) Toda system on a two-dimensional manifold M without boundary…”

The conformal plate buckling equation

“…For λ = 1 the fourth-order equation (1.1) is equivalent to the nonlinear system of second-order elliptic PDEs −∆u(x) = K(x)e 2u(x) , (1.4) −∆K(x) = e 2u(x) , (1.5) which describes a conformally flat surface over R 2 with metric g = e 2u g 0 and Gauss curvature function K ≡ K g generated in a self-consistent manner. While a considerably literature has accumulated about the celebrated prescribed Gauss curvature problem where K is given and only u has to be found by solving (1.4), see [2,3,5,6,8,10,11,12,13,14,15,16,17,23,26,27,30,31] and further references therein, the literature on selfconsistent Gauss curvature problems is relatively sparse [7,18,22,24]. In particular, we are not aware of any previous study of the self-consistent Gauss curvature problem (1.4), (1.5), equivalently the conformal plate buckling equation.…”