Vol. 8, No. 8, 2015

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

Aleks Jevnikar, Sadok Kallel and Andrea Malchiodi

Vol. 8 (2015), No. 8, 1963–2027

We consider the Toda system of Liouville equations on a compact surface Σ

Δu1 = 2ρ1( h1eu1 Σh1eu1dV g 1) ρ2( h2eu2 Σh2eu2dV g 1), Δu2 = 2ρ2( h2eu2 Σh2eu2dV g 1) ρ1( h1eu1 Σh1eu1dV g 1),

which arises as a model for nonabelian Chern–Simons vortices. Here h1 and h2 are smooth positive functions and ρ1 and ρ2 two positive parameters.

For the first time, the ranges ρ1 (4kπ,4(k + 1)π), k , and ρ2 (4π,8π) are studied with a variational approach on surfaces with arbitrary genus. We provide a general existence result by using a new improved Moser–Trudinger-type inequality and introducing a topological join construction in order to describe the interaction of the two components u1 and u2.

geometric PDEs, variational methods, min-max schemes
Mathematical Subject Classification 2010
Primary: 35J50, 35J61, 35R01
Received: 19 March 2015
Accepted: 7 September 2015
Published: 23 December 2015
Aleks Jevnikar
Scuola Internazionale Superiore di Studi Avanzati
Via Bonomea 265
I-34136 Trieste
Sadok Kallel
American University of Sharjah
University City
26666 Sharjah
United Arab Emirates
Laboratoire Painlevé
Université de Lille 1
Cité scientifique
Batiment M2
59655 Villeneuve d’Ascq
Andrea Malchiodi
Department of Mathematics
Scuola Normale Superiore
Piazza dei Cavalieri 7
I-50126 Pisa