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A topological join construction and the Toda system on compact surfaces of arbitrary genus. (English) Zbl 1336.35152

Authors’ abstract: We consider the Toda system of Liouville eqations on a compact surface \(\Sigma \) \[ \begin{cases} -\Delta u_1= 2\rho _1 \left( \frac{h_1e^{u_1}}{\int _{\Sigma } h_1 e^{u_1}dV_g} -1 \right) -\rho _2 \left( \frac{h_2 e^{u_2}}{\int _{\Sigma } h_2 e^{u_2}dV_g } -1\right), \\ -\Delta u_2= 2\rho _2 \left( \frac{h_2e^{u_2}}{\int _{\Sigma } h_2 e^{u_2}dV_g} -1 \right) -\rho _1 \left( \frac{h_1 e^{u_1}}{\int _{\Sigma } h_1 e^{u_1}dV_g } -1\right), \end{cases} \] 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 \(\rho _1 \in (4k \pi ,4(k+1)\pi ), k \in {\mathbb N}\), and \(\rho _2 \in (4\pi ,8\pi )\) 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 \(u_2\).

MSC:

35J50 Variational methods for elliptic systems
35J61 Semilinear elliptic equations
35R01 PDEs on manifolds
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