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

Alberto Farina; Alberto Ferrero

doi:10.1016/j.anihpc.2014.11.005

JournalsaihpcVol. 33, No. 2pp. 495–528

Existence and stability properties of entire solutions to the polyharmonic equation $(- Δ)^{m} u = e^{u}$ for any $m \geq 1$

Alberto Farina
Université de Picardie Jules Verne, Faculté des Sciences, Rue Saint-Leu 33, 80039 Amiens, France
Alberto Ferrero
Università del Piemonte Orientale “Amedeo Avogadro”, Dipartimento di Scienze e Innovazione Tecnologica, Viale Teresa Michel 11, 15121 Alessandria, Italy

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

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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 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.

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Alberto Farina, Alberto Ferrero, Existence and stability properties of entire solutions to the polyharmonic equation $(- Δ)^{m} u = e^{u}$ for any $m \geq 1$ . Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 2, pp. 495–528

DOI 10.1016/J.ANIHPC.2014.11.005