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) + u^av^b = 0 in Ω,
div(|∇v|^m-2&\ abla;v) + u^cv^d = 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.