We give a new approach on general systems of the form (G){[c]{c}%
-\Delta_{p}u=\operatorname{div}(|\nabla u| ^{p-2}\nabla u)=\epsilon_{1}|x|
^{a}u^{s}v^{\delta}, -\Delta_{q}v=\operatorname{div}(|\nabla v|^{q-2}\nabla
u)=\epsilon_{2}|x|^{b}u^{\mu}v^{m}, where Q,p,q,δ,μ,s,m,a,b are
real parameters, Q,p,q=1, and ϵ1=±1,ϵ2=±1. In
the radial case we reduce the problem to a quadratic system of order 4, of
Kolmogorov type. Then we obtain new local and global existence or nonexistence
results. In the case ϵ1=ϵ2=1, we also describe the
behaviour of the ground states in two cases where the system is variational. We
give an important result on existence of ground states for a nonvariational
system with p=q=2 and s=m>0. In the nonradial case we solve a conjecture of
nonexistence of ground states for the system with p=q=2 and δ=m+1 and
μ=s+1.Comment: 43 page