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
Abstract

We consider the Toda system of Liouville equations on a compact surface $\Sigma$

$\left\{\begin{array}{c}-\Delta {u}_{1}=2{\rho }_{1}\left(\frac{{h}_{1}{e}^{{u}_{1}}}{{\int }_{\Sigma }{h}_{1}{e}^{{u}_{1}}\phantom{\rule{0.3em}{0ex}}d{V}_{g}}-1\right)-{\rho }_{2}\left(\frac{{h}_{2}{e}^{{u}_{2}}}{{\int }_{\Sigma }{h}_{2}{e}^{{u}_{2}}\phantom{\rule{0.3em}{0ex}}d{V}_{g}}-1\right),\phantom{\rule{1em}{0ex}}\hfill \\ & \\ -\Delta {u}_{2}=2{\rho }_{2}\left(\frac{{h}_{2}{e}^{{u}_{2}}}{{\int }_{\Sigma }{h}_{2}{e}^{{u}_{2}}\phantom{\rule{0.3em}{0ex}}d{V}_{g}}-1\right)-{\rho }_{1}\left(\frac{{h}_{1}{e}^{{u}_{1}}}{{\int }_{\Sigma }{h}_{1}{e}^{{u}_{1}}\phantom{\rule{0.3em}{0ex}}d{V}_{g}}-1\right),\phantom{\rule{1em}{0ex}}\hfill \end{array}\right\$

which arises as a model for nonabelian Chern–Simons vortices. Here ${h}_{1}$ and ${h}_{2}$ are smooth positive functions and ${\rho }_{1}$ and ${\rho }_{2}$ two positive parameters.

For the first time, the ranges ${\rho }_{1}\in \left(4k\pi ,4\left(k+1\right)\pi \right)$, $k\in ℕ$, and ${\rho }_{2}\in \left(4\pi ,8\pi \right)$ 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 ${u}_{1}$ and ${u}_{2}$.

Keywords
geometric PDEs, variational methods, min-max schemes
Mathematical Subject Classification 2010
Primary: 35J50, 35J61, 35R01