Liouville-type theorems and decay estimates for solutions to higher order elliptic equations

Lu, Guozhen ; Wang, Peiyong ; Zhu, Jiuyi

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012), p. 653-665 / Harvested from Numdam

Résumé

Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Poláčik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations.

Détail

Publié le : 2012-01-01

MR 2971025 | Zbl 1255.35064

DOI : https://doi.org/10.1016/j.anihpc.2012.02.004
Classification:  35B53,  35J40,  35J47,  35B45

Bibliographie

[1] S. Agmon, A. Douglis, L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equation satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959), 623-729 | MR 125307 | Zbl 0093.10401

[2] H. Berestycki, L. Caffarelli, L. Nirenberg, Further qualitative properties for elliptic equation in unbounded domains, Ann. Sc. Norm. Super. Pisa Cl. Sci. 25 (1997), 69-94 | Numdam | MR 1655510 | Zbl 1079.35513

[3] I. Birindelli, E. Mitidieri, Liouville theorems for elliptic inequalities and applications, Proc. Roy. Soc. Edinburgh Sect. A 128 (1998), 1217-1247 | MR 1664101 | Zbl 0919.35023

[4] J. Busca, R. Manasevich, A Liouville-type theorem for Lane–Emden systems, Indiana Univ. Math. J. 51 no. 1 (2002), 37-51 | MR 1896155 | Zbl 1033.35032

[5] 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

[6] W. Chen, C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J. 63 (1991), 615-622 | MR 1121147 | Zbl 0768.35025

[7] W. Chen, C. Li, B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math. 59 no. 3 (2006), 330-343 | MR 2200258 | Zbl 1093.45001

[8] E. Colorado, I. Peral, Semilinear elliptic problems with mixed Dirichlet–Neumann boundary conditions, J. Funct. Anal. 199 (2003), 468-507 | MR 1971262 | Zbl 1034.35041

[9] E. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc. 46 (1992), 425-434 | MR 1190345

[10] L. Damascelli, F. Gladiali, Some nonexistence results for positive solutions of elliptic equations in unbounded domains, Rev. Mat. Iberoamericana 20 (2004), 67-86 | MR 2076772 | Zbl 1330.35146

[11] D.G. De Figueiredo, P.L. Lions, R.D. Nussbaum, A priori estimate and existence of positive solution to semilinear elliptic equations, J. Math. Pures Appl. 61 (1982), 41-63 | MR 664341 | Zbl 0452.35030

[12] D. De Figueiredo, B. Sirakov, Liouville type theorem, monotonicity results and a priori bounds for positive solution of elliptic systems, Math. Ann. 33 no. 2 (2005), 231-260 | MR 2195114 | Zbl 1165.35360

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

[14] B. Gidas, W. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 no. 3 (1979), 209-243 | MR 544879 | Zbl 0425.35020

[15] B. Gidas, J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), 525-598 | MR 615628 | Zbl 0465.35003

[16] B. Gidas, J. Spruck, A priori bounds for positive solutions for nonlinear elliptic equations, Comm. Partial Differential Equations 6 (1981), 883-901 | MR 619749 | Zbl 0462.35041

[17] G. Lu, J. Zhu, The axial symmetry and regularity of solutions to an integral equation in a half space, Pacific J. Math. 253 no. 2 (2011), 455-473 | MR 2878819 | Zbl 1239.45005

[18] E. Mitidieri, Nonexistence of positive solutions of semilinear elliptic systems in $ {\mathbb{R}}^{N}$, Differential Integral Equations 9 (1996), 465-479 | MR 1371702 | Zbl 0848.35034

[19] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 no. 1–2 (1993), 125-151 | MR 1211727 | Zbl 0816.35027

[20] P. Poláčik, P. Quittner, P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, Duke Math. J. 139 (2007), 555-579 | MR 2350853 | Zbl 1146.35038

[21] 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

[22] J. Serrin, H. Zou, Non-existence of positive solutions of Lane–Emden systems, Differential Integral Equations 9 no. 4 (1996), 635-653 | MR 1401429 | Zbl 0868.35032

[23] B. Sirakov, Existence results and a priori bounds for higher order elliptic equations and systems, J. Math. Pures Appl. 89 (2008), 114-133 | MR 2391643 | Zbl 1180.35214

[24] P. Souplet, The proof of the Lane–Emden conjecture in four space dimensions, Adv. Math. 221 no. 5 (2009), 1409-1427 | MR 2522424 | Zbl 1171.35035

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