Nonlinear fractional and singular systems: Nonexistence, existence, uniqueness, and Hölder regularity

摘要

In the present paper, we investigate the following singular quasilinear elliptic system: S −Δp1s1u=1uα1vβ1,u>0inΩ;u=0,inℝNΩ,−Δp2s2v=1vα2uβ2,v>0inΩ;v=0,inℝNΩ, where Ω⊂ℝN is an open‐bounded domain with smooth boundary, s1, s2 ∈ (0, 1),  p1, p2 ∈ (1, + ∞), and α1, α2, β1, β2 are positive constants. We first discuss the nonexistence of positive classical solutions to system (S). Next, constructing suitable ordered pairs of subsolutions and supersolutions, we apply Schauder's fixed‐point theorem in the associated conical shell and get the existence of a positive weak solutions pair to (S), turn to be Hölder continuous. Finally, we apply a well‐known Krasnosel'skiı̆'s argument to establish the uniqueness of such positive pair of solutions.