Sharp Liouville results for fully nonlinear equations with power-growth nonlinearities

Armstrong, Scott N.¹ ; Sirakov, Boyan²

¹ Department of Mathematics       Louisiana State University Baton Rouge, LA 70803
² UFR SEGMI Université Paris 10 92001 Nanterre Cedex, France and CAMS, EHESS 54 bd Raspail 75270 Paris Cedex 06, France

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 729-746.

Résumé

We study fully nonlinear elliptic equations such as

$$F\left(D^{2}u\right)=u^p,p>1,$$

in ${ℝ}^n$ 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 ${ℝ}^n$, 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.

Détail

Publié le : 2018-06-21

Classification : 35B53, 35J60

Bibliographie

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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. | MR | Zbl | EuDML | mathdoc-id

[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 | Zbl

[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. | MR | Zbl | EuDML | mathdoc-id

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl