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.
Keywords: Higher order equations, Radial solutions, Stability properties, Hardy–Rellich inequalities
@article{AIHPC_2016__33_2_495_0,
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},
year = {2016},
publisher = {Elsevier},
volume = {33},
number = {2},
doi = {10.1016/j.anihpc.2014.11.005},
zbl = {1336.35033},
mrnumber = {3465384},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.11.005/}
}
TY - JOUR AU - Farina, Alberto AU - Ferrero, Alberto TI - Existence and stability properties of entire solutions to the polyharmonic equation (−Δ) m u = e u for any m ≥1 JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 495 EP - 528 VL - 33 IS - 2 PB - Elsevier UR - http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.11.005/ DO - 10.1016/j.anihpc.2014.11.005 LA - en ID - AIHPC_2016__33_2_495_0 ER -
%0 Journal Article %A Farina, Alberto %A Ferrero, Alberto %T Existence and stability properties of entire solutions to the polyharmonic equation (−Δ) m u = e u for any m ≥1 %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 495-528 %V 33 %N 2 %I Elsevier %U http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2014.11.005/ %R 10.1016/j.anihpc.2014.11.005 %G en %F AIHPC_2016__33_2_495_0
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
Cité par Sources :