论文标题
在尺寸二维系统中，对泳道系统的积极解决方案的渐近行为二
Asymptotic behavior of positive solutions to the Lane-Emden system in dimension two
论文作者
论文摘要
考虑车道填充系统 \ 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*}，其中$ω$是$ \ mathbb {r}^n $中的平滑界域，带有$ n \ geq 2 $和$ q \ ge p> 0 $。该系统的{\至少能量解决方案}的渐近行为是以$ n \ geq 3 $进行了研究的。但是，$ n = 2 $的情况是不同的，并且保持完全开放。在本文中，我们研究了案例$ n = 2 $ $ q = p+θ_p$和$ \sup_pθ_p<+\ infty $。在以下自然条件下，为最少的能源解决方案持有 $ \ limsup_ {p \ to+\ infty} p \int_Ω\ nabla u_p \ cdot \ cdot \ nabla v_p \ nabla v_p \ mathrm {d} x <+\ infty，$ $我们完全描述{\ it itementions} $ _p，v _e $（I. $ p \ to+\ iy $。这似乎是在二维情况下泳道系统渐近行为的第一个结果。
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 $Ω$ 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 $q=p+θ_p$ and $\sup_pθ_p<+\infty$. Under the following natural condition that holds for least energy solutions $$\limsup_{p\to+\infty} p\int_Ω\nabla u_p\cdot\nabla v_p \mathrm{d} x<+\infty,$$ 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.
