Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, p. 729-746

We study fully nonlinear elliptic equations such as

F(D2u)=up,p>1,

in n or in exterior domains, where F is any uniformly elliptic, positively homogeneous operator. We show that there exists a critical exponent, depending on the homogeneity of the fundamental solution of F, that sharply characterizes the range of p>1 for which there exist positive supersolutions or solutions in any exterior domain. Our result generalizes theorems of Bidaut-Véron [6] as well as Cutri and Leoni [11], who found critical exponents for supersolutions in the whole space n , in case -F is Laplace’s operator and Pucci’s operator, respectively. The arguments we present are new and rely only on the scaling properties of the equation and the maximum principle.

Published online : 2018-06-21
Classification:  35B53,  35J60
@article{ASNSP_2011_5_10_3_729_0,
     author = {Armstrong, Scott N. and Sirakov, Boyan},
     title = {Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {3},
     year = {2011},
     pages = {729-746},
     zbl = {1250.35050},
     mrnumber = {2905384},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_729_0}
}
Armstrong, Scott N.; Sirakov, Boyan. Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, pp. 729-746. http://www.numdam.org/item/ASNSP_2011_5_10_3_729_0/

[1] H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 (1976), 620–709. | MR 415432 | Zbl 0345.47044

[2] S. N. Armstrong, Principal eigenvalues and an anti-maximum principle for homogeneous fully nonlinear elliptic equations, J. Differential Equations 246 (2009), 2958–2987. | MR 2503031 | Zbl 1171.35093

[3] S. N. Armstrong, B. Sirakov and C. K. Smart, Fundamental solutions of homogeneous fully nonlinear elliptic equations, Comm. Pure Appl. Math. 64 (2011), 737–777. | MR 2663711 | Zbl 1233.35089

[4] T. B. Benjamin, A unified theory of conjugate flows, Philos. Trans. Roy. Soc. London Ser. A 269 (1971), 587–643. | MR 446075 | Zbl 0226.76037

[5] M.-F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math. 84 (2001), 1–49. | MR 1849197 | Zbl 1018.35040

[6] M.-F. Bidaut-Véron, Local and global behavior of solutions of quasilinear equations of Emden-Fowler type, Arch. Rational Mech. Anal. 107 (1989), 293–324. | MR 1004713 | Zbl 0696.35022

[7] X. Cabré, Elliptic PDE’s in probability and geometry: symmetry and regularity of solutions, Discrete Contin. Dyn. Syst. 20 (2008), 425–457. | MR 2373200 | Zbl 1158.35033

[8] L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiȩch, On viscosity solutions of fully nonlinear equations with measurable ingredients, Comm. Pure Appl. Math. 49 (1996), 365–397. | MR 1376656 | Zbl 0854.35032

[9] L. A. Caffarelli and X. Cabré, “Fully Nonlinear Elliptic Equations”, Vol. 43 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 1995. | MR 1351007 | Zbl 0834.35002

[10] M. G. Crandall, Hitoshi Ishii and P.-L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1–67. | MR 1118699 | Zbl 0755.35015

[11] A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000),219–245. | Numdam | MR 1753094 | Zbl 0956.35035

[12] D. G. de Figueiredo, P.-L. Lions and R. D. Nussbaum, A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl. (9) 61 (1982), 41–63. | MR 664341 | Zbl 0452.35030

[13] P. L. Felmer and A. Quaas, On critical exponents for the Pucci’s extremal operators, Ann. Inst. H. Poincaré Anal. Non Linéaire 20 (2003),843–865. | Numdam | MR 1995504 | Zbl 1274.35115

[14] P. L. Felmer and A. Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc. 361 (2009), 5721–5736. | MR 2529911 | Zbl 1181.35090

[15] W. H. Fleming and H. M. Soner, “Controlled Markov Processes and Viscosity Solutions”, Vol. 25 of Stochastic Modelling and Applied Probability. Springer, New York, second edition, 2006. | MR 2179357 | Zbl 1105.60005

[16] B. Gidas and 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

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

[18] M. Guedda and L. Véron, Local and global properties of solutions of quasilinear elliptic equations, J. Differential Equations 76 (1988), 159–189. | MR 964617 | Zbl 0661.35029

[19] M. A. Krasnosel’skiǐ and P. P. Zabreǐko, “Geometrical Methods of Nonlinear Analysis”, Vol. 263 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer-Verlag, Berlin, 1984. Translated from the Russian by Christian C. Fenske. | MR 736839 | Zbl 0546.47030

[20] N. V. Krylov, “Nonlinear Elliptic and Parabolic Equations of the Second Order”, Vol. 7 of Mathematics and its Applications (Soviet Series), D. Reidel Publishing Co., Dordrecht, 1987. | MR 901759 | Zbl 0619.35004

[21] D. A. Labutin, Isolated singularities for fully nonlinear elliptic equations, J. Differential Equations 177 (2001), 49–76. | MR 1867613 | Zbl 0998.35017

[22] E. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001), 1–384. | MR 1879326 | Zbl 1074.35500

[23] A. Quaas and B. Sirakov, Existence results for nonproper elliptic equations involving the Pucci operator, Comm. Partial Differential Equations 31 (2006), 987–1003. | MR 2254600 | Zbl 1237.35056

[24] A. Quaas and B. Sirakov, Principal eigenvalues and the Dirichlet problem for fully nonlinear elliptic operators, Adv. Math. 218 (2008), 105–135. | MR 2409410 | Zbl 1143.35077

[25] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math. 189 (2002), 79–142. | MR 1946918 | Zbl 1059.35040