A new dynamical approach of Emden-Fowler equations and systems

Abstract

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 $\mu=s+1.$