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 & Continuous Dynamical Systems, 2016, 36 (9) : 4703-4721. doi: 10.3934/dcds.2016004
References:
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[2]

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

O. González-Meléndez and A. Quaas, On critical exponents for Lane-Emden-Fowler type equations with a singular extremal operator,, Submitted for publication., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

P. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Elect. J. Diff. Eqns., 2001 (2001), 1-19.  Google Scholar

[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.  Google Scholar

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), 758-780. doi: 10.1016/j.jde.2015.09.004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

O. González-Meléndez and A. Quaas, On critical exponents for Lane-Emden-Fowler type equations with a singular extremal operator,, Submitted for publication., ().   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[31]

O. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York-London, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

P. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Elect. J. Diff. Eqns., 2001 (2001), 1-19.  Google Scholar

[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.  Google Scholar

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