In this paper we develop a comprehensive study on principal eigenvalues and maximum and comparison principles related to the nonlocal Lane–Emden problem

where \(\Omega \) is a smooth bounded open subset of \({\mathbb {R}}^n\) with \(n \ge 1\), \(s,t\in (0,1)\), \(\alpha , \beta > 0\) satisfy \(\alpha \beta =1\), \(\rho \) and \(\tau \) are positive continuous functions on \(\Omega \) and \((-\Delta )^{s}\) and \((-\Delta )^{t}\) stand for fractional Laplace operators with powers s and t, respectively. By mean of topological arguments, sub-supersolution method and maximum principles to nonlocal elliptic operators, we show that the set of principal eigenvalues \((\lambda ,\mu )\) of the above problem is nonempty and in addition can be parameterized by a curve located in the first quadrant of the cartesian plane which satisfies some properties as continuity, simplicity, local isolation, monotonicity and also asymptotes on the coordinates axes. Moreover, its components can be represented through a min–max type type formula. Using some of these properties, we characterize all couples \((\lambda , \mu ) \in {\mathbb {R}}^2\) such that (weak and strong) maximum and comparison principles associated to the above problem holds in \(\Omega \). As a byproduct, we derive results on existence and uniqueness of viscosity solution for fractional elliptic systems on bounded domains with sublinear behavior.

在本文中，我们对与非局部Lane-Emden问题有关的主要特征值和最大值以及比较原理进行了全面研究

其中\（\ Omega \）是\（{\ mathbb {R}} ^ n \）具有\（n \ ge 1 \），\（s，t \ in（0,1）\ ），\（\ alpha，\ beta> 0 \）满足\（\ alpha \ beta = 1 \），\（\ rho \）和\（\ tau \）是\（\ Omega \）和\（（-\ Delta）^ {s} \）和\（（-\ Delta）^ {t} \）代表幂为s和t的分数阶拉普拉斯算子， 分别。通过拓扑论证，子超解法和非局部椭圆算子的最大原理，我们证明上述问题的本征值\（（\ lambda，\ mu）\）的集合是非空的，并且可以通过位于笛卡尔平面第一象限的曲线，满足一些特性，如连续性，简单性，局部隔离性，单调性以及坐标轴上的渐近性。而且，它的组成可以通过最小-最大类型类型公式来表示。利用这些属性中的某些，我们对所有对\（（\ lambda，\ mu）\ in {\ mathbb {R}} ^ 2 \）进行特征化，使得与上述问题相关的（弱和强）最大值和比较原理成立\（\ Omega \）。作为副产品，我们得出具有次线性行为的有界域上分数椭圆系统的粘性解的存在性和唯一性的结果。