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.

Abstract (VO)
Résumé

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 $R^{n}$ .

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 $R^{n}$ .

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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     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},
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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/

Bibliographie
Cité par

[1] W. Chen; C. Li Classification of solutions of some nonlinear elliptic equations, Duke Math. J., Volume 63 (1991), pp. 615-622

[2] W. Chen; C. Li Super polyharmonic property of solutions for PDE systems and its applications, Commun. Pure Appl. Anal., Volume 12 (2013), pp. 2497-2514

[3] W. Chen; C. Li; B. Ou Classification of solutions for an integral equation, Commun. Pure Appl. Math., Volume 59 (2006), pp. 330-343

[4] W. Chen; Y. Fang; C. Li Super poly-harmonic property of solutions for Navier boundary problems on a half space, J. Funct. Anal., Volume 265 (2013), pp. 1522-1555

[5] Y.S. Choi; X. Xu Nonlinear biharmonic equations with negative exponents, J. Differ. Equ., Volume 246 (2009), pp. 216-234

[6] T.V. Duoc; Q.A. Ngo On radial solutions of $Δ^{2} u + u^{- q} = 0$ in $R^{3}$ with exactly quadratic growth at infinity, Differ. Integral Equ. (2017) (in press) | arXiv

[7] Y. Fang; W. Chen A Liouville type theorem for poly-harmonic Dirichlet problems in a half space, Adv. Math., Volume 229 (2012), pp. 2835-2867

[8] A. Farina; A. Ferrero Existence and stability properties of entire solutions to the polyharmonic equation ${(- Δ)}^{m} u = e^{u}$ for any $m ⩾ 1$ , Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016), pp. 495-528

[9] I. Guerra A note on nonlinear biharmonic equations with negative exponents, J. Differ. Equ., Volume 253 (2012), pp. 3147-3157

[10] Z.M. Guo; J.C. Wei Entire solutions and global bifurcations for a biharmonic equation with singular non-linearity in $R^{3}$ , Adv. Differ. Equ., Volume 13 (2008), pp. 753-780

[11] B. Lai; D. Ye Remarks on entire solutions for two fourth-order elliptic problems, Proc. Edinb. Math. Soc., Volume 59 (2016), pp. 777-786

[12] J. Liu; Y. Guo; Y. Zhang Liouville-type theorems for polyharmonic systems in $R^{N}$ , J. Differ. Equ., Volume 225 (2006), pp. 685-709

[13] L. Ma; J. Wei Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal., Volume 254 (2008), pp. 1058-1087

[14] L. Martinazzi Classification of solutions to the higher order Liouville's equation on $R^{2 m}$ , Math. Z., Volume 263 (2009), pp. 307-329

[15] P.J. McKenna; W. Reichel Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differ. Equ., Volume 37 (2003), pp. 1-13

[16] J. Wei; X. Xu Classification of solutions of higher order conformally invariant equations, Math. Ann., Volume 313 (1999), pp. 207-228

[17] X. Xu Uniqueness theorem for the entire positive solutions of biharmonic equations in $R^{n}$ , Proc. R. Soc. Edinb., Sect. A, Math., Volume 130 (2000), pp. 651-670

[18] X. Xu Exact solutions of nonlinear conformally invariant integral equations in $R^{3}$ , Adv. Math., Volume 194 (2005), pp. 485-503

Wenxiong Chen; Wei Dai; Guolin Qin Liouville type theorems, a priori estimates and existence of solutions for critical and super-critical order Hardy-Hénon type equations in $R^{n}$ , Mathematische Zeitschrift, Volume 303 (2023) no. 4, p. 36 (Id/No 104) | DOI:10.1007/s00209-023-03265-y | Zbl:1519.35043
Wei Dai; Jingze Fu On properties of positive solutions to nonlinear tri-harmonic and bi-harmonic equations with negative exponents, Bulletin of Mathematical Sciences, Volume 12 (2022) no. 3, p. 48 (Id/No 2250007) | DOI:10.1142/s1664360722500072 | Zbl:1505.35128
Marius Ghergu; Yasuhito Miyamoto; Vitaly Moroz Polyharmonic inequalities with nonlocal terms, Journal of Differential Equations, Volume 296 (2021), pp. 799-821 | DOI:10.1016/j.jde.2021.06.019 | Zbl:1468.35054
Quc Anh Ngô; Van Hoang Nguyen; Quoc Hung Phan; Dong Ye Exhaustive existence and non-existence results for some prototype polyharmonic equations in the whole space, Journal of Differential Equations, Volume 269 (2020) no. 12, pp. 11621-11645 | DOI:10.1016/j.jde.2020.07.041 | Zbl:1454.35104
Quôc Anh Ngô Classification of entire solutions of $(- Δ)^{N} u + u^{- (4 N - 1)} = 0$ with exact linear growth at infinity in ${{\mathbb}} {R}^{2N-1}$ , Proceedings of the American Mathematical Society, Volume 146 (2018) no. 6, pp. 2585-2600 | DOI:10.1090/proc/13960 | Zbl:1391.35025

Cité par 5 documents. Sources : zbMATH

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