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

Abstract

Consider the Lane-Emden system \begin{equation*}\begin{aligned} &-Δu=v^p,\quad u>0,\quad\text{in}~Ω, &-Δv=u^q,\quad v>0,\quad\text{in}~Ω, &u=v=0,\quad\text{on}~\partialΩ, \end{aligned}\end{equation*} where $\Omega$ is a smooth bounded domain in ${\mathbb{R}}^N$ with $N\ge 2$ and $q\ge p>0$. The asymptotic behavior of {\it least energy solutions} of this system was studied for $N\ge 3$. 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 $$\underset{p\to +\infty }{\mathrm{limsup}}p{\int }_{\Omega }\nabla u_p\cdot \nabla v_p\mathrm{d}x<+\infty ,$$ 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.