# American Institute of Mathematical Sciences

September  2016, 36(9): 4703-4721. doi: 10.3934/dcds.2016004

## Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

 1 Departamento de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna, Spain 2 Dpto. de Análisis Matemático, Universidad de La Laguna, C/. Astrofísico Francisco Sánchez s/n, 38271 - La Laguna 3 Departamento de Matemática, Universidad Técnico Fedrico Santa María, Casilla V-110, Avda. España, 1680 - Valparaíso, Chile

Received  May 2015 Revised  January 2016 Published  May 2016

In this paper we consider positive supersolutions of the elliptic equation $-\triangle u = f(u) |\nabla u|^q$, posed in exterior domains of $\mathbb{R}^N$ ($N\ge 2$), where $f$ is continuous in $[0,+\infty)$ and positive in $(0,+\infty)$ and $q>0$. We classify supersolutions $u$ into four types depending on the function $m(R)=\inf_{|x|=R} u(x)$ for large $R$, and give necessary and sufficient conditions in order to have supersolutions of each of these types. As a consequence, we also obtain Liouville theorems for supersolutions depending on the values of $N$, $q$ and on some integrability properties on $f$ at zero or infinity. We also describe these questions when the equation is posed in the whole $\mathbb{R}^N$.
Citation: M. Á. Burgos-Pérez, J. García-Melián, A. Quaas. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete and Continuous Dynamical Systems, 2016, 36 (9) : 4703-4721. doi: 10.3934/dcds.2016004
##### References:
 [1] S. Alarcón, M. Burgos-Pérez, J. García Melián and A. Quaas, Nonexistence results for elliptic equations with gradient terms, Differential Equations, 260 (2016), 758-780. doi: 10.1016/j.jde.2015.09.004. [2] S. Alarcón, J. García-Melián and A. Quaas, Nonexistence of positive supersolutions to some nonlinear elliptic problems, J. Math. Pures Appl., 99 (2013), 618-634. doi: 10.1016/j.matpur.2012.10.001. [3] S. Alarcón, J. García Melián and A. Quaas, Liouville type theorems for elliptic equations with gradient terms, Milan J. Math., 81 (2013), 171-185. doi: 10.1007/s00032-013-0197-z. [4] S. Alarcón, J. García-Melián and A. Quaas, Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian, Ann. Scuola Norm. Sup. Pisa., 16 (2016), 129-158. doi: 10.2422/2036-2145.201402\_007. [5] S. N. Armstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Part. Diff. Eqns., 36 (2011), 2011-2047. doi: 10.1080/03605302.2010.534523. [6] S. N. Armstrong and B. Sirakov, Liouville results for fully nonlinear elliptic equations with power growth nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 711-728. [7] C. Bandle and M. Essén, On positive solutions of Emden equations in cone-like domains, Arch. Rational Mech. Anal., 112 (1990), 319-338. doi: 10.1007/BF02384077. [8] C. Bandle and H. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., 316 (1989), 595-622. doi: 10.1090/S0002-9947-1989-0937878-9. [9] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78. [10] 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. doi: 10.1007/BF00251552. [11] M. F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49. doi: 10.1007/BF02788105. [12] I. Birindelli and F. Demengel, Some Liouville theorems for the $p$-Laplacian, 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and Systems. Electr. J. Diff. Eqns. Conf., 8 (2002), 35-46. [13] I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse, 13 (2004), 261-287. doi: 10.5802/afst.1070. [14] L. Boccardo, F. Murat and J. P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa, 11 (1984), 213-235. [15] I. Capuzzo Dolcetta and A. Cutrì, Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Commun. Contemp. Math., 5 (2003), 435-448. doi: 10.1142/S0219199703001014. [16] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [17] M. Chipot and F. B. Weissler, Some blow-up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907. doi: 10.1137/0520060. [18] A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré (C) An. Non Linéaire, 17 (2000), 219-245. doi: 10.1016/S0294-1449(00)00109-8. [19] P. Felmer and A.Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (2009), 5721-5736. doi: 10.1090/S0002-9947-09-04566-8. [20] P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023. [21] R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916. doi: 10.1016/j.na.2008.12.018. [22] E. I. Galakhov, Solvability of an elliptic equation with a gradient nonlinearity, Differential Equations, 41 (2005), 693-702; Translated from Differentsial'nye Uravneniya, 41 (2005), 661-669. doi: 10.1007/s10625-005-0204-4. [23] E. I. Galakhov, Positive solutions of quasilinear elliptic equations, Math. Notes, 78 (2005), 185-193; Translated from Matematicheskie Zametki, 78 (2005), 202-211. doi: 10.1007/s11006-005-0114-z. [24] B. Gidas, Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, In Nonlinear partial differential equations in engineering and applied science, volume 54 of Lecture Notes in Pure and Appl. Math., pages 255-273. Marcel Dekker, New York, 1980. [25] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. [26] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0. [27] O. González-Meléndez and A. Quaas, On critical exponents for Lane-Emden-Fowler type equations with a singular extremal operator, Submitted for publication. [28] V. Kondratiev, V. Liskevich and V. Moroz, Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 25-43. doi: 10.1016/j.anihpc.2004.03.003. [29] V. Kondratiev, V. Liskevich and Z. Sobol, Positive solutions to semi-linear and quasi-linear elliptic equations on unbounded domains, In Handbook of differential equations: Stationary partial differential equations, Elsevier, 6 (2008), 177-267. doi: 10.1016/S1874-5733(08)80020-4. [30] V. Kondratiev, V. Liskevich and Z. Sobol, Positive supersolutions to semi-linear second-order non-divergence type elliptic equations in exterior domains, Trans. Amer. Math. Soc., 361 (2009), 697-713. doi: 10.1090/S0002-9947-08-04453-X. [31] O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. [32] Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87. doi: 10.1007/BF02786551. [33] V. Liskevich, I. I. Skrypnik and I. V. Skrypnik, Positive supersolutions to general nonlinear elliptic equations in exterior domains, Manuscripta Math., 115 (2004), 521-538. doi: 10.1007/s00229-004-0514-5. [34] J. Serrin and H. Zou, Existence and non-existence results for ground states of quasi-linear elliptic equations, Arch. Rat. Mech. Anal., 121 (1992), 101-130. doi: 10.1007/BF00375415. [35] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645. [36] P. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Elect. J. Diff. Eqns., 2001 (2001), 1-19. [37] F. X. Voirol, Coexistence of singular and regular solutions for the equation of Chipot and Weissler, Acta Math. Univ. Comenianae, 65 (1996), 53-64.

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##### References:
 [1] S. Alarcón, M. Burgos-Pérez, J. García Melián and A. Quaas, Nonexistence results for elliptic equations with gradient terms, Differential Equations, 260 (2016), 758-780. doi: 10.1016/j.jde.2015.09.004. [2] S. Alarcón, J. García-Melián and A. Quaas, Nonexistence of positive supersolutions to some nonlinear elliptic problems, J. Math. Pures Appl., 99 (2013), 618-634. doi: 10.1016/j.matpur.2012.10.001. [3] S. Alarcón, J. García Melián and A. Quaas, Liouville type theorems for elliptic equations with gradient terms, Milan J. Math., 81 (2013), 171-185. doi: 10.1007/s00032-013-0197-z. [4] S. Alarcón, J. García-Melián and A. Quaas, Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian, Ann. Scuola Norm. Sup. Pisa., 16 (2016), 129-158. doi: 10.2422/2036-2145.201402\_007. [5] S. N. Armstrong and B. Sirakov, Nonexistence of positive supersolutions of elliptic equations via the maximum principle, Comm. Part. Diff. Eqns., 36 (2011), 2011-2047. doi: 10.1080/03605302.2010.534523. [6] S. N. Armstrong and B. Sirakov, Liouville results for fully nonlinear elliptic equations with power growth nonlinearities, Ann. Sc. Norm. Super. Pisa Cl. Sci., 10 (2011), 711-728. [7] C. Bandle and M. Essén, On positive solutions of Emden equations in cone-like domains, Arch. Rational Mech. Anal., 112 (1990), 319-338. doi: 10.1007/BF02384077. [8] C. Bandle and H. Levine, On the existence and nonexistence of global solutions of reaction-diffusion equations in sectorial domains, Trans. Amer. Math. Soc., 316 (1989), 595-622. doi: 10.1090/S0002-9947-1989-0937878-9. [9] H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg, Superlinear indefinite elliptic problems and nonlinear Liouville theorems, Topol. Methods Nonlinear Anal., 4 (1994), 59-78. [10] 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. doi: 10.1007/BF00251552. [11] M. F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49. doi: 10.1007/BF02788105. [12] I. Birindelli and F. Demengel, Some Liouville theorems for the $p$-Laplacian, 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and Systems. Electr. J. Diff. Eqns. Conf., 8 (2002), 35-46. [13] I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators, Ann. Fac. Sci. Toulouse, 13 (2004), 261-287. doi: 10.5802/afst.1070. [14] L. Boccardo, F. Murat and J. P. Puel, Résultats d'existence pour certains problèmes elliptiques quasilinéaires, Ann. Scuola Norm. Sup. Pisa, 11 (1984), 213-235. [15] I. Capuzzo Dolcetta and A. Cutrì, Hadamard and Liouville type results for fully nonlinear partial differential inequalities, Commun. Contemp. Math., 5 (2003), 435-448. doi: 10.1142/S0219199703001014. [16] W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622. doi: 10.1215/S0012-7094-91-06325-8. [17] M. Chipot and F. B. Weissler, Some blow-up results for a nonlinear parabolic equation with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907. doi: 10.1137/0520060. [18] A. Cutrì and F. Leoni, On the Liouville property for fully nonlinear equations, Ann. Inst. H. Poincaré (C) An. Non Linéaire, 17 (2000), 219-245. doi: 10.1016/S0294-1449(00)00109-8. [19] P. Felmer and A.Quaas, Fundamental solutions and two properties of elliptic maximal and minimal operators, Trans. Amer. Math. Soc., 361 (2009), 5721-5736. doi: 10.1090/S0002-9947-09-04566-8. [20] P. Felmer and A. Quaas, Fundamental solutions and Liouville type theorems for nonlinear integral operators, Adv. Math., 226 (2011), 2712-2738. doi: 10.1016/j.aim.2010.09.023. [21] R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916. doi: 10.1016/j.na.2008.12.018. [22] E. I. Galakhov, Solvability of an elliptic equation with a gradient nonlinearity, Differential Equations, 41 (2005), 693-702; Translated from Differentsial'nye Uravneniya, 41 (2005), 661-669. doi: 10.1007/s10625-005-0204-4. [23] E. I. Galakhov, Positive solutions of quasilinear elliptic equations, Math. Notes, 78 (2005), 185-193; Translated from Matematicheskie Zametki, 78 (2005), 202-211. doi: 10.1007/s11006-005-0114-z. [24] B. Gidas, Symmetry properties and isolated singularities of positive solutions of nonlinear elliptic equations, In Nonlinear partial differential equations in engineering and applied science, volume 54 of Lecture Notes in Pure and Appl. Math., pages 255-273. Marcel Dekker, New York, 1980. [25] B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598. doi: 10.1002/cpa.3160340406. [26] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983. doi: 10.1007/978-3-642-61798-0. [27] O. González-Meléndez and A. Quaas, On critical exponents for Lane-Emden-Fowler type equations with a singular extremal operator, Submitted for publication. [28] V. Kondratiev, V. Liskevich and V. Moroz, Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 25-43. doi: 10.1016/j.anihpc.2004.03.003. [29] V. Kondratiev, V. Liskevich and Z. Sobol, Positive solutions to semi-linear and quasi-linear elliptic equations on unbounded domains, In Handbook of differential equations: Stationary partial differential equations, Elsevier, 6 (2008), 177-267. doi: 10.1016/S1874-5733(08)80020-4. [30] V. Kondratiev, V. Liskevich and Z. Sobol, Positive supersolutions to semi-linear second-order non-divergence type elliptic equations in exterior domains, Trans. Amer. Math. Soc., 361 (2009), 697-713. doi: 10.1090/S0002-9947-08-04453-X. [31] O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968. [32] Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations, J. Anal. Math., 90 (2003), 27-87. doi: 10.1007/BF02786551. [33] V. Liskevich, I. I. Skrypnik and I. V. Skrypnik, Positive supersolutions to general nonlinear elliptic equations in exterior domains, Manuscripta Math., 115 (2004), 521-538. doi: 10.1007/s00229-004-0514-5. [34] J. Serrin and H. Zou, Existence and non-existence results for ground states of quasi-linear elliptic equations, Arch. Rat. Mech. Anal., 121 (1992), 101-130. doi: 10.1007/BF00375415. [35] J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645. [36] P. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Elect. J. Diff. Eqns., 2001 (2001), 1-19. [37] F. X. Voirol, Coexistence of singular and regular solutions for the equation of Chipot and Weissler, Acta Math. Univ. Comenianae, 65 (1996), 53-64.
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