On the sub poly-harmonic property for solutions to (−Δ)pu < 0 in $ {\mathbb{R}}^{n}$

Quốc Anh Ngô

doi:10.1016/j.crma.2017.04.003

Partial differential equations

On the sub poly-harmonic property for solutions to (−Δ)^pu < 0 in

R^{n}

[De la sous-poly-harmonicité des solutions de (−Δ)^pu < 0 dans

R^{n}

]

Présenté par : Haïm Brézis

Quốc Anh Ngô ¹

¹ Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viet Nam

Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 526-532.

Résumé
Abstract

Dans cette Note, nous étudions principalement la relation entre le signe de ${(- Δ)}^{p} u$ et ${(- Δ)}^{p - i} u$ dans $R^{n}$ pour $1 \leq i \leq p - 1$ , avec n, $p \geq 2$ . Étant donnée l'inégalité différentielle ${(- Δ)}^{p} < 0$ , nous montrons, dans un premier temps, plusieurs conditions suffisantes afin que l'inégalité ${(- Δ)}^{p - 1} u < 0$ soit satisfaite. Puis, sous une hypothèse de croissance, nous montrons que ${(- Δ)}^{i} u < 0$ pour tout $i = 1, 2, \dots, p - 1$ , c'est-à-dire que u satisfait la propriété de sous-poly-harmonicité. Dans la dernière partie de la Note, nous considérons la sur-poly-harmonicité des solutions de l'équation ${(- Δ)}^{p} u = e^{2 p u}$ et ${(- Δ)}^{p} u = u^{q}$ , avec $q > 0$ , dans $R^{n}$ .

In this note, we mainly study the relation between the sign of ${(- Δ)}^{p} u$ and ${(- Δ)}^{p - i} u$ in $R^{n}$ with $p ⩾ 2$ and $n ⩾ 2$ for $1 ⩽ i ⩽ p - 1$ . Given the differential inequality ${(- Δ)}^{p} u < 0$ , first we provide several sufficient conditions so that ${(- Δ)}^{p - 1} u < 0$ holds. Then we provide conditions such that ${(- Δ)}^{i} u < 0$ for all $i = 1, 2, \dots, p - 1$ , which is known as the sub poly-harmonic property for u. In the last part of the note, we revisit the super poly-harmonic property for solutions to ${(- Δ)}^{p} u = e^{2 p u}$ and ${(- Δ)}^{p} u = u^{q}$ with $q > 0$ in $R^{n}$ .

Reçu le : 2016-11-10
Accepté le : 2017-04-06
Publié le : 2017-04-21

DOI : 10.1016/j.crma.2017.04.003

Affiliations des auteurs :

Quốc Anh Ngô ¹

¹ Department of Mathematics, College of Science, Viêt Nam National University, Hà Nôi, Viet Nam

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     author = {Quốc Anh Ng\^o},
     title = {On the sub poly-harmonic property for solutions to {(\ensuremath{-}\ensuremath{\Delta})\protect\textsuperscript{\protect\emph{p}}\protect\emph{u}\,<\,0} in $ {\mathbb{R}}^{n}$},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {526--532},
     publisher = {Elsevier},
     volume = {355},
     number = {5},
     year = {2017},
     doi = {10.1016/j.crma.2017.04.003},
     language = {en},
}

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Quốc Anh Ngô. On the sub poly-harmonic property for solutions to (−Δ)^pu < 0 in $ {\mathbb{R}}^{n}$. Comptes Rendus. Mathématique, Volume 355 (2017) no. 5, pp. 526-532. doi : 10.1016/j.crma.2017.04.003. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.04.003/

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