Pointwise bounds and blow-up for nonlinear polyharmonic inequalities

Taliaferro, Steven D.

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 1069-1096 / Harvested from Numdam

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Résumé
Abstract

Nous obtenons des résultats pour la question suivante, avec $m ⩾ 1$ et $n ⩾ 2$ entiers. QuestionPour quelles fonctions continues $f : [0, \infty) \to [0, \infty)$ existe-t-il une fonction continue $ϕ : (0, 1) \to (0, \infty)$ telle que chaque solution $C^{2 m}$ non-negative $u (x)$ de $0 ⩽ - Δ^{m} u ⩽ f (u) dans B_{2} (0) ∖ {0} \subset ℝ^{n}$ satisfasse à $u (x) = O (ϕ(| x |)) lorsque x \to 0,$ et quelle est la meilleure de ces fonctions φ quand elle existe ?

We obtain results for the following question where $m ⩾ 1$ and $n ⩾ 2$ are integers. QuestionFor which continuous functions $f : [0, \infty) \to [0, \infty)$ does there exist a continuous function $ϕ : (0, 1) \to (0, \infty)$ such that every $C^{2 m}$ nonnegative solution $u (x)$ of $0 ⩽ - Δ^{m} u ⩽ f (u) in B_{2} (0) ∖ {0} \subset ℝ^{n}$ satisfies $u (x) = O (ϕ(| x |)) as x \to 0$ and what is the optimal such φ when one exists?

Publié le : 2013-01-01

MR 3132417 | Zbl 1286.35278

DOI : https://doi.org/10.1016/j.anihpc.2012.12.011
Classification: 35B09, 35B33, 35B40, 35B44, 35B45, 35R45, 35J30, 35J91

@article{AIHPC_2013__30_6_1069_0,
     author = {Taliaferro, Steven D.},
     title = {Pointwise bounds and blow-up for nonlinear polyharmonic inequalities},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     volume = {30},
     year = {2013},
     pages = {1069-1096},
     doi = {10.1016/j.anihpc.2012.12.011},
     mrnumber = {3132417},
     zbl = {1286.35278},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIHPC_2013__30_6_1069_0}
}

Taliaferro, Steven D. Pointwise bounds and blow-up for nonlinear polyharmonic inequalities. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) pp. 1069-1096. doi : 10.1016/j.anihpc.2012.12.011. http://gdmltest.u-ga.fr/item/AIHPC_2013__30_6_1069_0/

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