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November  2011, 10(6): 1747-1762. doi: 10.3934/cpaa.2011.10.1747

A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities

Bernd Kawohl 1, and  Vasilii Kurta 2,

1. 

Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany
2. 

Mathematical Reviews, 416 Fourth Street, P.O. Box 8604, Ann Arbor, Michigan 48107-8604, United States

Received  March 2011 Revised  April 2011 Published  May 2011

We compare entire weak solutions $u$ and $v$ of quasilinear partial differential inequalities on $R^n$ without any assumptions on their behaviour at infinity and show among other things, that they must coincide if they are ordered, i.e. if they satisfy $u\geq v$ in $R^n$. For the particular case that $v\equiv 0$ we recover some known Liouville type results. Model cases for the equations involve the $p$-Laplacian operator for $p\in[1,2]$ and the mean curvature operator.
Keywords: Liouville theorem, singular elliptic equation., comparison principle, quasilinear elliptic equation.
Mathematics Subject Classification: Primary: 35B53 35J62; Secondary: 35B51, 35J75, 35J92, 35J9.
Citation: Bernd Kawohl, Vasilii Kurta. A Liouville comparison principle for solutions of singular quasilinear elliptic second-order partial differential inequalities. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1747-1762. doi: 10.3934/cpaa.2011.10.1747
References:
[1]

I. Birindelli and F. Demengel, Some Liouville theorems for the p-Laplacian, Proceedings of the 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, 35-46 (electronic), Electron. J. Differ. Equ. Conf., 8, Southwest Texas State Univ., San Marcos, TX, 2002.  Google Scholar

[2]

H. Brezis, Semilinear equations in $R^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.  Google Scholar

[3]

L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type, Ann. Inst. H. Poincaré Anal. Non Linaire, 26 (2009), 1099-1119.  Google Scholar

[4]

L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities, Nonlinear Anal., 70 (2009), 2882-2888.  Google Scholar

[5]

A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, J. Differ. Eqs., 250 (2011), 4367-4436. Google Scholar

[6]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  Google Scholar

[7]

J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations," The Clarendon Press, Oxford University Press, New York, 1993.  Google Scholar

[8]

A. G. Kartsatos and R. D. Mabry, Controlling the space with preassigned responses, J. Optim. Theory Appl., 54 (1987), 517-540.  Google Scholar

[9]

A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1970, 403 p.  Google Scholar

[10]

V. A. Kondrat$'$ev and E. M. Landis, Semilinear second-order equations with nonnegative characteristic form, Mat. Zametki, 44 (1988), 457-468 (Russian).  Google Scholar

[11]

V. V. Kurta, Qualitative properties of solutions of some classes of second-order quasilinear elliptic equations, Differentsial$'$nye Uravneniya, 28 (1992), 867-873 (Russian).  Google Scholar

[12]

V. V. Kurta, "Some Problems of Qualitative Theory for Nonlinear Second-order Equations," Doctoral Dissert., Steklov Math. Inst., Moscow, 1994 (Russian). Google Scholar

[13]

V. V. Kurta, On the comparison principle for second-order quasilinear elliptic equations, Differentsial$'$nye Uravneniya, 31 (1995), 289-295 (Russian).  Google Scholar

[14]

V. V. Kurta, Comparison principle for solutions of parabolic inequalities, C. R. Acad. Sci. Paris, Série I, 322 (1996), 1175-1180.  Google Scholar

[15]

V. V. Kurta, Comparison principle and analogues of the Phragmén-Lindelöf theorem for solutions of parabolic inequalities, Appl. Anal., 71 (1999), 301-324.  Google Scholar

[16]

V. V. Kurta, On the absence of positive solutions of elliptic equations, Mat. Zametki, 65 (1999), 552-561; translation in Math. Notes, 65 (1999), 462-469 (Russian).  Google Scholar

[17]

J.-L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[18]

V. M. Miklyukov, A new approach to the Bernstein theorem and to related questions of equations of minimal surface type, Mat. Sb. (N.S.), 108 (150) (1979), 268-289 (Russian).  Google Scholar

[19]

E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, Dokl. Akad. Nauk, 359 (1998), 456-460 (Russian).  Google Scholar

[20]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  Google Scholar

[21]

J. Serrin, Entire solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 352 (2009), 3-14.  Google Scholar

show all references

References:
[1]

I. Birindelli and F. Demengel, Some Liouville theorems for the p-Laplacian, Proceedings of the 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and System, 35-46 (electronic), Electron. J. Differ. Equ. Conf., 8, Southwest Texas State Univ., San Marcos, TX, 2002.  Google Scholar

[2]

H. Brezis, Semilinear equations in $R^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.  Google Scholar

[3]

L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type, Ann. Inst. H. Poincaré Anal. Non Linaire, 26 (2009), 1099-1119.  Google Scholar

[4]

L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities, Nonlinear Anal., 70 (2009), 2882-2888.  Google Scholar

[5]

A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, J. Differ. Eqs., 250 (2011), 4367-4436. Google Scholar

[6]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  Google Scholar

[7]

J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations," The Clarendon Press, Oxford University Press, New York, 1993.  Google Scholar

[8]

A. G. Kartsatos and R. D. Mabry, Controlling the space with preassigned responses, J. Optim. Theory Appl., 54 (1987), 517-540.  Google Scholar

[9]

A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1970, 403 p.  Google Scholar

[10]

V. A. Kondrat$'$ev and E. M. Landis, Semilinear second-order equations with nonnegative characteristic form, Mat. Zametki, 44 (1988), 457-468 (Russian).  Google Scholar

[11]

V. V. Kurta, Qualitative properties of solutions of some classes of second-order quasilinear elliptic equations, Differentsial$'$nye Uravneniya, 28 (1992), 867-873 (Russian).  Google Scholar

[12]

V. V. Kurta, "Some Problems of Qualitative Theory for Nonlinear Second-order Equations," Doctoral Dissert., Steklov Math. Inst., Moscow, 1994 (Russian). Google Scholar

[13]

V. V. Kurta, On the comparison principle for second-order quasilinear elliptic equations, Differentsial$'$nye Uravneniya, 31 (1995), 289-295 (Russian).  Google Scholar

[14]

V. V. Kurta, Comparison principle for solutions of parabolic inequalities, C. R. Acad. Sci. Paris, Série I, 322 (1996), 1175-1180.  Google Scholar

[15]

V. V. Kurta, Comparison principle and analogues of the Phragmén-Lindelöf theorem for solutions of parabolic inequalities, Appl. Anal., 71 (1999), 301-324.  Google Scholar

[16]

V. V. Kurta, On the absence of positive solutions of elliptic equations, Mat. Zametki, 65 (1999), 552-561; translation in Math. Notes, 65 (1999), 462-469 (Russian).  Google Scholar

[17]

J.-L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[18]

V. M. Miklyukov, A new approach to the Bernstein theorem and to related questions of equations of minimal surface type, Mat. Sb. (N.S.), 108 (150) (1979), 268-289 (Russian).  Google Scholar

[19]

E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, Dokl. Akad. Nauk, 359 (1998), 456-460 (Russian).  Google Scholar

[20]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  Google Scholar

[21]

J. Serrin, Entire solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 352 (2009), 3-14.  Google Scholar

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