A remark on the generalized second order Toda system
Tóm tắt
In this paper, we establish a priori estimates to the generalized second order Toda system
$$\left\{ \begin{gathered}
- \Delta u_1 (x) = 2R_1 (x)e^{u_1 } - R_2 e^{u_2 } , \hfill \\
- \Delta u_2 (x) = - R_1 (x)e^{u_1 } + 2R_2 e^{u_2 } \hfill \\
\end{gathered} \right.$$
in ℝ2, and discuss the convergence and asymptotic behavior of its solutions, where R
i
(x), i = 1, 2, is bounded function in ℝ2. Consequently, we prove that all the solutions satisfy an identity, which is somewhat a generalization of the well-known Kazdan-Warner condition.
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