A topological join construction and the Toda system on compact surfaces of arbitrary genus

Abstract

We consider the following Toda system of Liouville equations on a compact surface Σ:    −∆u 1 = 2ρ 1 h 1 e u 1 Σ h 1 e u 1 dVg − 1 − ρ 2 h 2 e u 2 Σ h 2 e u 2 dVg − 1 , −∆u 2 = 2ρ 2 h 2 e u 2 Σ h 2 e u 2 dVg − 1 − ρ 1 h 1 e u 1 Σ h 1 e u 1 dVg − 1 , which arises as a model for non-abelian Chern-Simons vortices. Here h 1 , h 2 are smooth positive functions and ρ 1 , ρ 2 two positive parameters.For the first time the range ρ 1 ∈ (4kπ, 4(k + 1)π), k ∈ N, ρ 2 ∈ (4π, 8π) is studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by means of a new improved Moser-Trudinger type inequality and introducing a topological join construction in order to describe the interaction of the two components u 1 and u 2 .2000 Mathematics Subject Classification. 35J50, 35J61, 35R01. Key words and phrases. Geometric PDEs, Variational Methods, Min-max Schemes. A.J. and A.M. have been supported by the PRIN Variational and perturbative aspects of nonlinear differential problems. A.J. and A.M acknowledge support from the Mathematics Department at the University of Warwick. 1 arXiv:1503.05524v2 [math.AP] 19 Mar 2015 and (a, b 1This construction allows to map low sublevels of J ρ into Σ k * Σ l , with the join parameter s expressing whether distributionally e u1 is closer to Σ k or e u2 is closer to Σ l . The hypothesis on the genus of Σ in [3] was used in the following way: on such surfaces one can construct two disjoint simple non-contractible curves γ 1 , γ 2 such that Σ retracts on each of them through continuous maps Π 1 , Π 2 . By means of these retractions, low energy sublevels may be described in terms of (γ 1 ) k or (γ 2 ) l only. On the other hand, one can build test functions modelled on (γ 1 ) k * (γ 2 ) l for which each component u i only concentrates near γ i , to somehow minimize the interaction between the two components u 1 and u 2 , due to the fact that γ 1 and γ 2 are disjoint.We prove here the following result, which for the first time applies to surfaces of arbitrary genus when both parameters ρ i are supercritical and one of them also arbitrarily large.Theorem 1.1. Let h 1 , h 2 be two positive smooth functions and let Σ be any compact surface. Suppose that ρ 1 ∈ (4kπ, 4(k + 1)π), k ∈ N and ρ 2 ∈ (4π, 8π). Then problem (1) has a solution.Remark 1.2. Theorem 1.1 is new when Σ is a sphere and k ≥ 3. As we already discussed, the case of surfaces with positive genus was covered in [3]. The case of Σ S 2 and k = 1 was covered in [36], while for k = 2 it was covered in [31]. In the latter paper the authors indeed computed the Leray-Schauder degree of the equation for the range of ρ i 's in Theorem 1.1. It turns out that the degree of (1) is zero for the sphere when k ≥ 3: since solutions do exist by Theorem 1.1, it means that either they are degenerate, or that degrees of multiple ones cancel, so a global degree counting does not detect them. A similar phenomenon occurs for (5) on the sphere, when ρ > 12π, see [12]. Even for positive genus, we believe that our approach co...

Min–max schemes for SU(3) Toda systems

“…Then system (2) is solvable. 24,37]). Let h 1 , h 2 be two positive smooth functions and let Σ be any compact surface.…”

“…We will just sketch the main ideas needed to prove this result, referring to Section 3 in [24] for details. Under the assumptions of Theorem 1.8, when k ≥ 2 we have that for low values of J ρ (u) either e u1 is concentrated near at most k points of Σ or e u2 is concentrated near a single point.…”

“…We consider now the assumptions of Theorem 1.8, and for simplicity here we limit ourselves to the case k = 1, referring to [24] for when k ≥ 2.…”

B 2 and G2 Toda systems on compact surfaces: A variational approach

“…A general existence result have been given in [10,9] for (6), as well as some existence results for (5) under an upper bound on one or both the ρ i (see [25,26,17]). Moreover, in [4] existence of solutions is shown under no assumptions on ρ but rather on the topology of Σ.…”

“…Such an issue was tackled in [26,17] for the case of A 2 Toda system. The results in these papers exploit the structure of A 2 matrix and cannot be extended to other systems.…”