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

Abstract

We study the prototype model of the boundary value problem where $\mathrm{\Omega }\subset {\mathbf{R}}^n$ ($n\ge 2$) is a connected smooth domain, and the exponents $m>1$ and $a,b,c,d\ge 0$ are non-negative numbers. Under appropriate conditions on the exponents $m$, $a$, $b$, $c$ and $d$, and on the domain $\mathrm{\Omega }$, a variety of results on a priori estimates, existence and non-existence of positive solutions have been established.