Consider the Lane-Emden system \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=v^p,\quad u>0,\quad \text {in}~\Omega ,\\ -\Delta v=u^q,\quad v>0,\quad \text {in}~\Omega ,\\ u=v=0,\quad \text {on}~\partial \Omega , \end{array}\right. } \end{aligned}\) where \(\Omega \) is a smooth bounded domain in \(\mathbb {R}^N\) with \(N\ge 2\) and \(q\ge p>0\) . The asymptotic behavior of least energy solutions of this system was studied by Guerra [23] and Choi-Kim [8] for \(N\ge 3\) , while 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 energy condition that holds automatically for \(\Omega \) being star-shaped (see Kamburov-Sirakov [25]) \(\begin{aligned} \limsup _{p\rightarrow +\infty } p\int _\Omega \nabla u_p\cdot \nabla v_p \mathrm dx<+\infty , \end{aligned}\) we give a complete description of the asymptotic behavior of positive solutions \((u_p,v_p)\) as \(p\rightarrow +\infty \) . This seems the first result for asymptotic behaviors of the Lane-Emden system in the two dimension case. In a sequel work [7], we will apply this aysmptotic result to prove the uniqueness of positive solutions for large p when \(\Omega \) is a convex domain.
