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 $Q,p,q,\delta,\mu,s,m,$ $a,b$ are real parameters, $Q,p,q\neq1,$ and $\epsilon_{ 1}=\pm1,$ $\epsilon_{ 2}=\pm1.$ 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 $\epsilon_{ 1}=\epsilon_{ 2}=1,$ 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 $\delta=m+1$ and $\mu=s+1.$
Categories: math.AP