×

Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems. (English) Zbl 1211.35101

Summary: Let \({\mathbf B}=B_1(0)\) be the unit ball in \(\mathbb R^n\) and \(r=|x|\). We study the poly-harmonic Dirichlet problem
\[ \begin{cases} (-\Delta)^mu=f(u) &\text{in }{\mathbf B},\\ u=\frac{\partial u}{\partial r}=\cdots= \frac{\partial^{m-1}u}{\partial r^{m-1}}=0 &\text{on }\partial{\mathbf B}. \end{cases} \]
Using the corresponding integral equation and the method of moving planes in integral forms, we show that the positive solutions are radially symmetric and monotone decreasing about the origin. We also obtain regularity for solutions.

MSC:

35J40 Boundary value problems for higher-order elliptic equations
35J30 Higher-order elliptic equations
35J35 Variational methods for higher-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I, Comm. pure appl. math., 12, 623-727, (1959) · Zbl 0093.10401
[2] Boggio, T., Sulle fuzioni di Green d’ordine m, Rend. circ. mat. Palermo, 20, 97-135, (1905) · JFM 36.0827.01
[3] Berchio, E.; Gazzola, F.; Weth, T., Radial symmetry of positive solutions to nonlinear polyharmonic Dirichlet problems, J. reine angew. math., 620, 165-183, (2008) · Zbl 1182.35109
[4] Chen, W.; Jin, C.; Li, C.; Lim, C., Weighted Hardy-Littlewood-Sobolev inequalities and system of integral equations, Discrete contin. dyn. syst. S, 164-172, (2005) · Zbl 1147.45301
[5] Chen, W.; Li, C., Regularity of solutions for a system of integral equations, Commun. pure appl. anal., 4, 1-8, (2005) · Zbl 1073.45004
[6] Chen, W.; Li, C., The best constant in some weighted Hardy-Littlewood-Sobolev inequality, Proc. amer. math. soc., 136, 955-962, (2008) · Zbl 1132.35031
[7] Chen, W.; Li, C., Classification of positive solutions for nonlinear differential and integral systems with critical exponents, Acta math. sci., 4, 949-960, (2009) · Zbl 1212.35103
[8] Chen, W.; Li, C., An integral system and the Lane-Emden conjecture, Discrete contin. dyn. syst., 4, 1167-1184, (2009) · Zbl 1176.35067
[9] Chen, W.; Li, C., Methods on nonlinear elliptic equations, AIMS ser. differ. equ. dyn. syst., vol. 4, (2010)
[10] W. Chen, C. Li, The equivalences between integral systems and PDE systems, preprint, 2010.
[11] Chen, W.; Li, C., A \(\sup + \inf\) inequality near \(R = 0\), Adv. math., 220, 219-245, (2009) · Zbl 1157.35364
[12] Chen, W.; Li, C.; Ou, B., Classification of solutions for an integral equation, Comm. pure appl. math., 59, 330-343, (2006) · Zbl 1093.45001
[13] Chen, W.; Li, C.; Ou, B., Qualitative properties of solutions for an integral equation, Discrete contin. dyn. syst., 12, 347-354, (2005) · Zbl 1081.45003
[14] Gidas, B.; Ni, W.; Nirenberg, L., Symmetry of related properties via the maximum principle, Comm. math. phys., 68, 209-243, (1979) · Zbl 0425.35020
[15] Hang, F., On the integral systems related to Hardy-Littlewood-Sobolev inequality, Math. res. lett., 14, 373-383, (2007) · Zbl 1144.26031
[16] Hang, F.; Wang, X.; Yan, X., An integral equation in conformal geometry, Ann. inst. H. Poincaré anal. non lineaire, 26, 1-21, (2009) · Zbl 1154.45004
[17] Jin, C.; Li, C., Symmetry of solutions to some systems of integral equations, Proc. amer. math. soc., 134, 1661-1670, (2006) · Zbl 1156.45300
[18] Li, Y., Remark on some conformally invariant integral equations: the method of moving spheres, J. eur. math. soc., 6, 153-180, (2004) · Zbl 1075.45006
[19] Lieb, E.; Loss, M., Analysis, (2001), Amer. Math. Soc. RI
[20] Li, C.; Ma, L., Uniqueness of positive bound states to Schrödinger systems with critical exponents, SIAM J. appl. anal., 40, 1049-1057, (2008) · Zbl 1167.35347
[21] Liu, C.; Qiao, S., Symmetry and monotonicity for a system of integral equations, Commun. pure appl. anal., 6, 1925-1932, (2009) · Zbl 1185.45011
[22] Li, D.; Zhuo, R., An integral equation on half space, Proc. amer. math. soc., 138, 2779-2791, (2010) · Zbl 1200.45001
[23] Ma, L.; Chen, D., A Liouville type theorem for an integral system, Commun. pure appl. anal., 5, 855-859, (2006) · Zbl 1134.45007
[24] Ma, L.; Chen, D., Radial symmetry and monotonicity results for an integral equation, J. math. anal. appl., 2, 943-949, (2008) · Zbl 1140.45004
[25] Ma, L.; Chen, D., Radial symmetry and uniqueness of non-negative solutions to an integral system, Math. comput. modelling, 49, 379-385, (2009)
[26] C. Ma, W. Chen, C. Li, Regularity of solutions for an integral system of Wolff type, Adv. Math., doi:10.1016/j.aim.2010.07.020. · Zbl 1209.45006
[27] Ma, L.; Zhao, L., Classification of positive solitary solutions of the nonlinear Choquard equation, Arch. ration. mech. anal., 2, 455-467, (2010) · Zbl 1185.35260
[28] Sweers, G., On gidas-ni-Nirenberg type result for biharmonic problems, Math. nachr., 246, 202-206, (2002) · Zbl 1158.35314
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.