论文标题
平面域中车道系统积极解决方案的尖锐估计，唯一性和非平稳性
Sharp estimates, uniqueness and nondegeneracy of positive solutions of the Lane-Emden system in planar domains
论文作者
论文摘要
我们研究车道填充系统$$ \ begin {cases}-Δu= v^p，\ quad u> 0，\ quad \ quad \ text {in} 〜Ω，-ΔV= u^q，\ quad v> 0，\ quad v> 0，\ quad \ quad \ quad \ text \ text {in}其中$ω\ subset \ mathbb {r}^2 $是一个平滑的有限域。在最近的一项工作中，我们研究了正解决方案的浓度现象为$ p，q \ to+\ infty $和$ | q-p | \ leqleqλ$。在本文中，我们获得了此类多泡解决方案的尖锐估计，包括局部最大值和缩放参数的清晰收敛速率，以及溶液的准确近似值。作为这些尖锐估计的应用，我们表明当$ω$是凸面时，该系统的解决方案是唯一的，对于$ p，q $的大型且非排效。
We study the Lane-Emden system $$\begin{cases} -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{cases}$$ where $Ω\subset\mathbb{R}^2$ is a smooth bounded domain. In a recent work, we studied the concentration phenomena of positive solutions as $p,q\to+\infty$ and $|q-p|\leq Λ$. In this paper, we obtain sharp estimates of such multi-bubble solutions, including sharp convergence rates of local maxima and scaling parameters, and accurate approximations of solutions. As an application of these sharp estimates, we show that when $Ω$ is convex, then the solution of this system is unique and nondegenerate for large $p, q$.
