A remark on the generalized second order Toda system (Q383619)

Let \(R_1\), \(R_2\in L^\infty(\Omega)\). In this paper, the following Toda system in \(\mathbb R^2\) \[ \begin{cases} -\Delta u_1(x)=2R_1(x)e^{u_1}-R_2(x)e^{u_2}\\ -\Delta u_2(x)=-R_1(x)e^{u_1}+2R_2(x)e^{u_2}\end{cases} \] is considered. For solutions \((u_1,u_2)\in H_{loc}^1(\mathbb R^2)\times H_{loc}^1(\mathbb R^2)\) satisfying the inequalities \(\int_{\mathbb R^2}e^{u_i}<\infty\), \(i=1,2\), the a-priori estimate \[ \sup_{x\in \mathbb R^2}\{u_1(x),u_2(x)\}<\infty \] is established. Moreover, for a bounded domain \(\Omega\) containing the origin and for \(R_1\), \(R_2\) positive in \(\Omega\) with \(R_1(x)\leq R_1(0)\), \(R_2(x)\leq R_2(0)\) a.e. \(x\in\Omega\), the author proves a convergence result for sequences \(\{(u_1^k,u_2^k)\}\) of solutions to the above system satisfying, for some constant \(C\), the inequalities \(\int_{\mathbb R^2}e^{u_i^k}\leq C\), for all \(i\in\{1,2\}\) and \(k\in \mathbb N\). Finally, for positive \(R_1\), \(R_2\), the author studies the asymptotic behavior of solutions \((u_1,u_2)\) satisfying the inequalities \(\int_{\mathbb R^2}e^{u_i(y)}(\ln(|y|+1)+1)dy<\infty\), \(i=1,2\).