Principal curves to nonlocal Lane–Emden systems and related maximum principles

Abstract

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

$$\begin{aligned} \left\{ \begin{array}{llll} (-\Delta )^{s}u = \lambda \rho (x)\vert v\vert ^{\alpha -1}v & \mathrm{in} \ \ \Omega ,\\ (-\Delta )^{t}v = \mu \tau (x)\vert u\vert ^{\beta -1}u & \mathrm{in} \ \ \Omega ,\\ u= v=0 & \mathrm{in} \ \ {\mathbb {R}}^n{\setminus }\Omega , \end{array} \right. \end{aligned}$$

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.