Pointwise bounds and blow-up for nonlinear polyharmonic inequalities
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 6, pp. 1069-1096.

Nous obtenons des résultats pour la question suivante, avec $m⩾1$ et $n⩾2$ entiers. QuestionPour quelles fonctions continues $f:\left[0,\infty \right)\to \left[0,\infty \right)$ existe-t-il une fonction continue $\varphi :\left(0,1\right)\to \left(0,\infty \right)$ telle que chaque solution ${C}^{2m}$ non-negative $u\left(x\right)$ de

 $0⩽-{\Delta }^{m}u⩽f\left(u\right)\phantom{\rule{1em}{0ex}}\text{dans}\phantom{\rule{0.166667em}{0ex}}{B}_{2}\left(0\right)\setminus \left\{0\right\}\subset {ℝ}^{n}$
satisfasse à
 $u\left(x\right)=O\left(\varphi \right(|x|\left)\right)\phantom{\rule{1em}{0ex}}\text{lorsque}\phantom{\rule{0.166667em}{0ex}}x\to 0,$
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 $f:\left[0,\infty \right)\to \left[0,\infty \right)$ does there exist a continuous function $\varphi :\left(0,1\right)\to \left(0,\infty \right)$ such that every ${C}^{2m}$ nonnegative solution $u\left(x\right)$ of

 $0⩽-{\Delta }^{m}u⩽f\left(u\right)\phantom{\rule{1em}{0ex}}\text{in}\phantom{\rule{0.166667em}{0ex}}{B}_{2}\left(0\right)\setminus \left\{0\right\}\subset {ℝ}^{n}$
satisfies
 $u\left(x\right)=O\left(\varphi \right(|x|\left)\right)\phantom{\rule{1em}{0ex}}\text{as}\phantom{\rule{0.166667em}{0ex}}x\to 0$
and what is the optimal such φ when one exists?

DOI : https://doi.org/10.1016/j.anihpc.2012.12.011
Classification : 35B09,  35B33,  35B40,  35B44,  35B45,  35R45,  35J30,  35J91
Mots clés : Isolated singularity, Polyharmonic, Blow-up, Pointwise bound
@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},
pages = {1069--1096},
publisher = {Elsevier},
volume = {30},
number = {6},
year = {2013},
doi = {10.1016/j.anihpc.2012.12.011},
zbl = {1286.35278},
mrnumber = {3132417},
language = {en},
url = {http://www.numdam.org/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) no. 6, pp. 1069-1096. doi : 10.1016/j.anihpc.2012.12.011. http://www.numdam.org/item/AIHPC_2013__30_6_1069_0/

[1] S. Armstrong, B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Partial Differential Equations 36 (2011), 2011-2047 | MR 2846170 | Zbl 1230.35030

[2] G. Caristi, L. DʼAmbrosio, E. Mitidieri, Representation formulae for solutions to some classes of higher order systems and related Liouville theorems, Milan J. Math. 76 (2008), 27-67 | MR 2465985 | Zbl 1186.35026

[3] G. Caristi, E. Mitidieri, R. Soranzo, Isolated singularities of polyharmonic equations, Atti Semin. Mat. Fis. Univ. Modena 46 (1998), 257-294 | MR 1645722 | Zbl 0915.35035

[4] Y.S. Choi, X. Xu, Nonlinear biharmonic equations with negative exponents, J. Differential Equations 246 (2009), 216-234 | MR 2467021 | Zbl 1165.35014

[5] T. Futamura, K. Kishi, Y. Mizuta, Removability of sets for sub-polyharmonic functions, Hiroshima Math. J. 33 (2003), 31-42 | MR 1966650 | Zbl 1041.31004

[6] T. Futamura, Y. Mizuta, Isolated singularities of super-polyharmonic functions, Hokkaido Math. J. 33 (2004), 675-695 | MR 2104835 | Zbl 1073.31005

[7] F. Gazzola, H.-C. Grunau, G. Sweers, Polyharmonic Boundary Value Problems, Springer (2010) | MR 2667016

[8] M. Ghergu, A. Moradifam, S.D. Taliaferro, Isolated singularities of polyharmonic inequalities, J. Funct. Anal. 261 (2011), 660-680 | MR 2799575 | Zbl 1218.35255

[9] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer (1983) | MR 737190 | Zbl 0691.35001

[10] Y. Guo, J. Liu, Liouville-type theorems for polyharmonic equations in ${ℝ}^{N}$ and in ${ℝ}_{+}^{N}$, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), 339-359 | MR 2406694 | Zbl 1153.35028

[11] S.-Y. Hsu, Removable singularity of the polyharmonic equation, Nonlinear Anal. 72 (2010), 624-627 | MR 2579330 | Zbl 1184.35008

[12] P.J. Mckenna, W. Reichel, Radial solutions of singular nonlinear biharmonic equations and applications to conformal geometry, Electron. J. Differential Equations 37 (2003) | EuDML 127118 | MR 1971023 | Zbl 1109.35321

[13] W. Reichel, T. Weth, A priori bounds and a Liouville theorem on a half-space for higher-order elliptic Dirichlet problems, Math. Z. 261 (2009), 805-827 | MR 2480759 | Zbl 1167.35014

[14] W. Reichel, T. Weth, Existence of solutions to nonlinear, subcritical higher order elliptic Dirichlet problems, J. Differential Equations 248 (2010), 1866-1878 | MR 2593610 | Zbl 1185.35066

[15] S.D. Taliaferro, On the growth of superharmonic functions near an isolated singularity II, Comm. Partial Differential Equations 26 (2001), 1003-1026 | MR 1843293 | Zbl 0979.31003

[16] S.D. Taliaferro, Isolated singularities on nonlinear elliptic inequalities, Indiana Univ. Math. J. 50 (2001), 1885-1897 | MR 1889086 | Zbl 1101.35323

[17] J. Wei, X. Xu, Classification of solutions of higher order conformally invariant equations, Math. Ann. 313 (1999), 207-228 | MR 1679783 | Zbl 0940.35082

[18] X. Xu, Uniqueness theorem for the entire positive solutions of biharmonic equations in ${R}^{n}$, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), 651-670 | MR 1769247 | Zbl 0961.35037