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.

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 ?

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?

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

[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 | Zbl

[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 | Zbl

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

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

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

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

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

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

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

[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 | Zbl

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

[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 | MR | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl