Existence and non-existence for strongly coupled quasi-linear cooperative elliptic systems

Abstract

We study the prototype model of the boundary value problem div(|▽u|m-2▽u) + uavb = 0 in Ω, div(|▽u|m-2▽v) + ucvd = 0 in Ω, u = v = 0 on ∂Ω, where Ω ⊂ Rn (n ≥ 2) is a connected smooth domain, and the exponents m > 1 and a,b,c,d ≥ 0 are non-negative numbers. Under appropriate conditions on the exponents m, a, b, c and d, and on the domain Ω, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established.