Existence and stability properties of entire solutions to the polyharmonic equation (−Δ) m u = e u for any m ≥1

Farina, Alberto; Ferrero, Alberto

doi:10.1016/j.anihpc.2014.11.005

Existence and stability properties of entire solutions to the polyharmonic equation (−Δ)^m u = e ^u for any m ≥1

Farina, Alberto ; Ferrero, Alberto

Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 495-528.

Résumé

We study existence and stability properties of entire solutions of a polyharmonic equation with an exponential nonlinearity. We study existence of radial entire solutions and we provide some asymptotic estimates on their behavior at infinity. As a first result on stability we prove that stable solutions (not necessarily radial) in dimensions lower than the conformal one never exist. On the other hand, we prove that radial entire solutions which are stable outside a compact set always exist both in high and low dimensions. In order to prove stability of solutions outside a compact set we prove some new Hardy–Rellich type inequalities in low dimensions.

MR Zbl

DOI : 10.1016/j.anihpc.2014.11.005

Classification : 35G20, 35B08, 35B35, 35B40
Mots clés : Higher order equations, Radial solutions, Stability properties, Hardy–Rellich inequalities

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     author = {Farina, Alberto and Ferrero, Alberto},
     title = {Existence and stability properties of entire solutions to the polyharmonic equation {(\ensuremath{-}\ensuremath{\Delta})\protect\textsuperscript{}            \protect\emph{m}         }         \protect\emph{u}         =         \protect\emph{e}         \protect\textsuperscript{            \protect\emph{u}         } for any \protect\emph{m}         \ensuremath{\geq}1},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {495--528},
     publisher = {Elsevier},
     volume = {33},
     number = {2},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.11.005},
     zbl = {1336.35033},
     mrnumber = {3465384},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2014.11.005/}
}

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Farina, Alberto; Ferrero, Alberto. Existence and stability properties of entire solutions to the polyharmonic equation (−Δ)^m         u         =         e         ^u for any m         ≥1. Annales de l'I.H.P. Analyse non linéaire, Tome 33 (2016) no. 2, pp. 495-528. doi : 10.1016/j.anihpc.2014.11.005. http://www.numdam.org/articles/10.1016/j.anihpc.2014.11.005/

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