Pointwise bounds and blow-up for nonlinear polyharmonic inequalities

Taliaferro, Steven D.

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 6, pp. 1069-1096

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 $\varphi :(0,1)\to (0,\infty )$ such that every $C^{2m}$ nonnegative solution $u\left(x\right)$ of

$$0⩽-{\Delta }^{m}u⩽f\left(u\right)\text{in}{B}_2\left(0\right)\setminus \left\{0\right\}\subset {ℝ}^n$$

satisfies

$$u\left(x\right)=O\left(\varphi \right(\left|x\right|\left)\right)\text{as}x\to 0$$

and what is the optimal such φ when one exists?

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 $\varphi :(0,1)\to (0,\infty )$ telle que chaque solution $C^{2m}$ non-negative $u\left(x\right)$ de

$$0⩽-{\Delta }^{m}u⩽f\left(u\right)\text{dans}{B}_2\left(0\right)\setminus \left\{0\right\}\subset {ℝ}^n$$

satisfasse à

$$u\left(x\right)=O\left(\varphi \right(\left|x\right|\left)\right)\text{lorsque}x\to 0,$$

et quelle est la meilleure de ces fonctions φ quand elle existe ?

Classification : 35B09, 35B33, 35B40, 35B44, 35B45, 35R45, 35J30, 35J91
Keywords: Isolated singularity, Polyharmonic, Blow-up, Pointwise bound

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