Asymptotic behavior of positive solutions to the Lane-Emden system in dimension two

Consider the Lane-Emden system \begin{equation*}\begin{aligned} &-\Delta u=v^p,\quad u>0,\quad\text{in}~\Omega, &-\Delta v=u^q,\quad v>0,\quad\text{in}~\Omega, &u=v=0,\quad\text{on}~\partial\Omega, \end{aligned}\end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$ with $N\geq 2$ and $q\ge p>0$. The asymptotic behavior of {\it least energy solutions} of this system was studied for $N\geq 3$. However, the case $N=2$ is different and remains completely open. In this paper, we study the case $N=2$ with $q=p+\theta_p$ and $\sup_p\theta_p<+\infty$. Under the following natural condition that holds for least energy solutions $$\limsup_{p\to+\infty} p\int_\Omega\nabla u_p\cdot\nabla v_p \mathrm{d} x<+\infty,$$ we give a complete description of the asymptotic behavior of {\it positive solutions} $(u_p,v_p)$ (i.e., not only for least energy solutions) as $p\to+\iy$. This seems the first result for asymptotic behaviors of the Lane-Emden system in the two dimension case.