A new dynamical approach of Emden-Fowler equations and systems

Abstract

We give a new approach to general Emden-Fowler equations and systems of the form \begin{equation*} (E_{\varepsilon })-\Delta _{p}u=-{\rm div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\varepsilon \left\vert x\right\vert ^{a}u^{Q}, \end{equation*} \begin{equation*} (G)\left\{ \begin{array}{c} -\Delta _{p}u=-{\rm div}(\left\vert \nabla u\right\vert ^{p-2}\nabla u)=\varepsilon _{1}\left\vert x\right\vert ^{a}u^{s}v^{\delta }, \\ -\Delta _{q}v=-{\rm div}(\left\vert \nabla v\right\vert ^{q-2}\nabla u)=\varepsilon _{2}\left\vert x\right\vert ^{b}u^{\mu }v^{m},% \end{array}% \right. \end{equation*}% where $p,q,Q,\delta, \mu, s,m,$ $a,b$ are real parameters, $p,q\neq 1,$ and $% \varepsilon, \varepsilon _{1},\varepsilon _{2}=\pm 1.$ In the radial case we reduce the problem $(G)$ to a quadratic system of four coupled first-order autonomous equations of Kolmogorov type. In the scalar case the two equations $(E_{\varepsilon })$ with source ($\varepsilon =1)$ or absorption (% $\varepsilon =-1)$ are reduced to a unique system of order 2. The reduction of system $(G)$ allows us to obtain new local and global existence or nonexistence results. We consider in particular the case $\varepsilon _{1}=\varepsilon _{2}=1.$ We describe the behaviour of the ground states when the system is variational. We give a result of existence of ground states for a nonvariational system with $p=q=2$ and $s=m>0,$ that improves the former ones. It is obtained by introducing a new type of energy function. In the nonradial case we solve a conjecture of nonexistence of ground states for the system with $p=q=2$, $\delta =m+1$ and $\mu =s+1.$