Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

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

Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

M. Á. Burgos-Pérez ^1, , J. García-Melián ^2, and A. Quaas ^3,

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

Abstract
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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 $\mathbb{R}^N$.

Keywords: classification of solutions., Nonlinear elliptic equations, Liouville theorems.

Mathematics Subject Classification: Primary: 35J15, 35J60; Secondary: 35B5.

Citation: M. Á. Burgos-Pérez, J. García-Melián, A. Quaas. Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems. Discrete & Continuous Dynamical Systems - A, 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. 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. 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. 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. 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. 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.
[7]	C. Bandle and M. Essén, On positive solutions of Emden equations in cone-like domains,, Arch. Rational Mech. Anal., 112* (1990), 319. 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. 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.
[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. 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. 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.
[13]	I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators,, Ann. Fac. Sci. Toulouse, 13 (2004), 261. 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.
[15]	I. Capuzzo Dolcetta and A. Cutrì, Hadamard and Liouville type results for fully nonlinear partial differential inequalities,, Commun. Contemp. Math., 5 (2003), 435. doi: 10.1142/S0219199703001014.
[16]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. 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. 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. 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. 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. doi: 10.1016/j.aim.2010.09.023.*
[21]	R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities,, Nonlinear Anal., 70 (2009), 2903. 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. doi: 10.1007/s10625-005-0204-4.
[23]	E. I. Galakhov, Positive solutions of quasilinear elliptic equations,, Math. Notes, 78 (2005), 185. 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, (1980), 255.
[25]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525. 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. 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, 6 (2008), 177. 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. doi: 10.1090/S0002-9947-08-04453-X.*
[31]	O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968).
[32]	Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations,, J. Anal. Math., 90 (2003), 27. 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. 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. 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. 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.*
[37]	F. X. Voirol, Coexistence of singular and regular solutions for the equation of Chipot and Weissler,, Acta Math. Univ. Comenianae, 65 (1996), 53.

show all references

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. 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. 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. 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. 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. 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.
[7]	C. Bandle and M. Essén, On positive solutions of Emden equations in cone-like domains,, Arch. Rational Mech. Anal., 112* (1990), 319. 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. 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.
[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. 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. 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.
[13]	I. Birindelli and F. Demengel, Comparison principle and Liouville type results for singular fully nonlinear operators,, Ann. Fac. Sci. Toulouse, 13 (2004), 261. 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.
[15]	I. Capuzzo Dolcetta and A. Cutrì, Hadamard and Liouville type results for fully nonlinear partial differential inequalities,, Commun. Contemp. Math., 5 (2003), 435. doi: 10.1142/S0219199703001014.
[16]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations,, Duke Math. J., 63 (1991), 615. 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. 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. 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. 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. doi: 10.1016/j.aim.2010.09.023.*
[21]	R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities,, Nonlinear Anal., 70 (2009), 2903. 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. doi: 10.1007/s10625-005-0204-4.
[23]	E. I. Galakhov, Positive solutions of quasilinear elliptic equations,, Math. Notes, 78 (2005), 185. 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, (1980), 255.
[25]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525. 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. 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, 6 (2008), 177. 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. doi: 10.1090/S0002-9947-08-04453-X.*
[31]	O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations,, Academic Press, (1968).
[32]	Y. Li and L. Zhang, Liouville-type theorems and Harnack-type inequalities for semilinear elliptic equations,, J. Anal. Math., 90 (2003), 27. 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. 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. 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. 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.*
[37]	F. X. Voirol, Coexistence of singular and regular solutions for the equation of Chipot and Weissler,, Acta Math. Univ. Comenianae, 65 (1996), 53.

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