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 a v b = 0 in Ω , div ( | ∇ v | m - 2 ∇ v ) + u c v d = 0 in Ω , u = v = 0 on ∂ Ω , where Ω ⊂ R n ( 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.