Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space
In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire positive solutions to the simplest models of polyharmonic equations with power-type nonlinearityΔmu=±uαin Rn with n⩾1, m⩾1, and α∈R. We aim to study the existence and non-existence of such classical solutio...
Gespeichert in:
Veröffentlicht in: | Journal of Differential Equations 2020-12, Vol.269 (12), p.11621-11645 |
---|---|
Hauptverfasser: | , , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
Zusammenfassung: | In this paper, we are interested in entire, non-trivial, non-negative solutions and/or entire positive solutions to the simplest models of polyharmonic equations with power-type nonlinearityΔmu=±uαin Rn with n⩾1, m⩾1, and α∈R. We aim to study the existence and non-existence of such classical solutions to the above equations in the full range of the constants n, m and α. Remarkably, we are able to provide necessary and sufficient conditions on the exponent α to guarantee the existence of such solutions in Rn. Finally, we identify all the situations where any entire non-trivial, non-negative classical solution must be positive everywhere. |
---|---|
ISSN: | 0022-0396 1090-2732 |
DOI: | 10.1016/j.jde.2020.07.041 |