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...

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Veröffentlicht in:Journal of Differential Equations 2020-12, Vol.269 (12), p.11621-11645
Hauptverfasser: Ngô, Quốc Anh, Nguyen, Van Hoang, Phan, Quoc Hung, Ye, Dong
Format: Artikel
Sprache:eng
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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