Existence and Multiplicity Results for Nonlocal Lane-Emden Systems

In this work, we show the existence and multiplicity for the nonlocal Lane-Emden system of the form

$$ \begin{array}{@{}rcl@{}} \left\{ \begin{aligned} \mathbb L u &= v^{p} + \rho \nu \quad &&\text{in } {\varOmega}, \\ \mathbb L v &= u^{q} + \sigma \tau \quad &&\text{in } {\varOmega},\\ u&=v = 0 \quad &&\text{on } \partial {\varOmega} \text{ or in } {\varOmega}^{c} \text{ if applicable}, \end{aligned} \right. \end{array} $$

where \({\varOmega } \subset \mathbb {R}^{N}\) is a C² bounded domain, \(\mathbb L\) is a nonlocal operator, ν,τ are Radon measures on Ω, p,q are positive exponents, and ρ,σ > 0 are positive parameters. Based on a fine analysis of the interaction between the Green kernel associated with \(\mathbb L\), the source terms u^q,v^p and the measure data, we prove the existence of a positive minimal solution. Furthermore, by analyzing the geometry of Palais-Smale sequences in finite dimensional spaces given by the Galerkin type approximations and their appropriate uniform estimates, we establish the existence of a second positive solution, under a smallness condition on the positive parameters ρ,σ and superlinear growth conditions on source terms. The contribution of the paper lies on our unifying technique that is applicable to various types of local and nonlocal operators.