November/December 2010 A new dynamical approach of Emden-Fowler equations and systems
Marie Françoise Bidaut-Veron, Hector Giacomini
Adv. Differential Equations 15(11/12): 1033-1082 (November/December 2010). DOI: 10.57262/ade/1355854434

## 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 $\mu =s+1.$

## Citation

Marie Françoise Bidaut-Veron. Hector Giacomini. "A new dynamical approach of Emden-Fowler equations and systems." Adv. Differential Equations 15 (11/12) 1033 - 1082, November/December 2010. https://doi.org/10.57262/ade/1355854434

## Information

Published: November/December 2010
First available in Project Euclid: 18 December 2012

zbMATH: 1230.34021
MathSciNet: MR2743494
Digital Object Identifier: 10.57262/ade/1355854434

Subjects:
Primary: 34B15 , 34C20 , 34C37 , 35J20 , 35J55 , 35J65 , 35J70 , 37J45

Rights: Copyright © 2010 Khayyam Publishing, Inc.

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