This paper focuses on maximum and comparison principles related to the Lane–Emden problem

$$\left\{\begin{array}{cc}-{\mathcal{L}}_{1}u=\lambda \rho (x){|v|}^{\alpha -1}v\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ -{\mathcal{L}}_{2}v=\mu \tau (x){|u|}^{\beta -1}u\hfill & \mathrm{in}\mathrm{\Omega },\hfill \\ u=v=0\hfill & \mathrm{on}\partial \mathrm{\Omega },\hfill \end{array}\right.$$

where $\alpha ,\beta >0$, $\alpha \beta =1$, $\mathrm{\Omega }$ is a smooth bounded open subset of ${\mathbb{R}}^n$ with $n⩾1$, $\rho$ and $\tau$ are positive functions on $\mathrm{\Omega }$ and ${\mathcal{L}}_1$ and ${\mathcal{L}}_2$ are second‐order uniformly elliptic linear operators in $\mathrm{\Omega }$. We characterize the couples $(\lambda ,\mu )\in {\mathbb{R}}^2$ such that the weak maximum principle associated to this problem holds in $\mathrm{\Omega }$. Moreover, weak and strong versions of maximum and comparison principles are also proved. As applications, we establish existence and uniqueness of solution for the non‐homogeneous counterpart of the above problem as well as Aleksandrov–Bakelman–Pucci‐type estimates. Lower bounds for principal eigenvalues of Lane–Emden systems in terms of the measure of $\mathrm{\Omega }$ are also derived.

中文翻译：

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Lane-Emden系统的最大值和比较原理

本文重点讨论与Lane-Emden问题有关的最大值和比较原理

$$\left\{\begin{array}{cc}-{\mathcal{大号}}_{\mathrm{1个}}ü=\lambda \rho （X）{|v|}^{\alpha -\mathrm{1个}}v\hfill & \mathrm{在}\mathrm{\Omega }，\hfill \\ -{\mathcal{大号}}_{2}v=\mu \tau （X）{|ü|}^{\beta -\mathrm{1个}}ü\hfill & \mathrm{在}\mathrm{\Omega }，\hfill \\ ü=v=0\hfill & \mathrm{上}\partial \mathrm{\Omega }，\hfill \end{array}\right.$$

哪里 $\alpha ，\beta >0$， $\alpha \beta =\mathrm{1个}$， $\mathrm{\Omega }$ 是...的一个光滑的有界开放子集 ${\mathbb{[R}}^{ñ}$ 与 $ñ⩾\mathrm{1个}$， $\rho$ 和 $\tau$ 对...有积极作用 $\mathrm{\Omega }$ 和 ${\mathcal{大号}}_{\mathrm{1个}}$ 和 ${\mathcal{大号}}_2$ 是二阶均匀椭圆线性算子 $\mathrm{\Omega }$。我们描绘了夫妻$（\lambda ，\mu ）\in {\mathbb{[R}}^2$ 这样，与此问题相关的弱最大原理就成立了 $\mathrm{\Omega }$。此外，还证明了最大值和比较原理的弱版本和强版本。作为应用，我们为上述问题的非均匀对应物以及Aleksandrov–Bakelman–Pucci型估计建立了解的存在性和唯一性。Lane-Emden系统主要特征值的下界，用$\mathrm{\Omega }$ 也得出。