Existence and stability properties of entire solutions to thepolyharmonic equation (-Δ)m u = eu for any m ≥ 1

Alberto Farina, Alberto Ferrero

Research output: Contribution to journalArticlepeer-review

Abstract

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 setalways 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.

Original languageEnglish
Pages (from-to)495-528
Number of pages34
JournalAnnales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume33
Issue number2
DOIs
Publication statusPublished - 1 Mar 2016

Keywords

  • Hardy-Rellich inequalities
  • Higher order equations
  • Radial solutions
  • Stability properties

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