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

Bernd Kawohl; Vasilii Kurta

doi:10.3934/cpaa.2011.10.1747

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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

Abstract
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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 and 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.
[2]	H. Brezis, Semilinear equations in $R^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.
[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.
[4]	L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities, Nonlinear Anal., 70 (2009), 2882-2888.
[5]	A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, J. Differ. Eqs., 250 (2011), 4367-4436.
[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.
[7]	J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations," The Clarendon Press, Oxford University Press, New York, 1993.
[8]	A. G. Kartsatos and R. D. Mabry, Controlling the space with preassigned responses, J. Optim. Theory Appl., 54 (1987), 517-540.
[9]	A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1970, 403 p.
[10]	V. A. Kondrat$'$ev and E. M. Landis, Semilinear second-order equations with nonnegative characteristic form, Mat. Zametki, 44 (1988), 457-468 (Russian).
[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).
[12]	V. V. Kurta, "Some Problems of Qualitative Theory for Nonlinear Second-order Equations," Doctoral Dissert., Steklov Math. Inst., Moscow, 1994 (Russian).
[13]	V. V. Kurta, On the comparison principle for second-order quasilinear elliptic equations, Differentsial$'$nye Uravneniya, 31 (1995), 289-295 (Russian).
[14]	V. V. Kurta, Comparison principle for solutions of parabolic inequalities, C. R. Acad. Sci. Paris, Série I, 322 (1996), 1175-1180.
[15]	V. V. Kurta, Comparison principle and analogues of the Phragmén-Lindelöf theorem for solutions of parabolic inequalities, Appl. Anal., 71 (1999), 301-324.
[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).
[17]	J.-L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Gauthier-Villars, Paris, 1969.
[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).
[19]	E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, Dokl. Akad. Nauk, 359 (1998), 456-460 (Russian).
[20]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.
[21]	J. Serrin, Entire solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 352 (2009), 3-14.

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.
[2]	H. Brezis, Semilinear equations in $R^N$ without condition at infinity, Appl. Math. Optim., 12 (1984), 271-282.
[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.
[4]	L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities, Nonlinear Anal., 70 (2009), 2882-2888.
[5]	A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, J. Differ. Eqs., 250 (2011), 4367-4436.
[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.
[7]	J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations," The Clarendon Press, Oxford University Press, New York, 1993.
[8]	A. G. Kartsatos and R. D. Mabry, Controlling the space with preassigned responses, J. Optim. Theory Appl., 54 (1987), 517-540.
[9]	A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis," Prentice-Hall, Inc., Englewood Cliffs, N.J., 1970, 403 p.
[10]	V. A. Kondrat$'$ev and E. M. Landis, Semilinear second-order equations with nonnegative characteristic form, Mat. Zametki, 44 (1988), 457-468 (Russian).
[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).
[12]	V. V. Kurta, "Some Problems of Qualitative Theory for Nonlinear Second-order Equations," Doctoral Dissert., Steklov Math. Inst., Moscow, 1994 (Russian).
[13]	V. V. Kurta, On the comparison principle for second-order quasilinear elliptic equations, Differentsial$'$nye Uravneniya, 31 (1995), 289-295 (Russian).
[14]	V. V. Kurta, Comparison principle for solutions of parabolic inequalities, C. R. Acad. Sci. Paris, Série I, 322 (1996), 1175-1180.
[15]	V. V. Kurta, Comparison principle and analogues of the Phragmén-Lindelöf theorem for solutions of parabolic inequalities, Appl. Anal., 71 (1999), 301-324.
[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).
[17]	J.-L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires," Dunod, Gauthier-Villars, Paris, 1969.
[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).
[19]	E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities, Dokl. Akad. Nauk, 359 (1998), 456-460 (Russian).
[20]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.
[21]	J. Serrin, Entire solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 352 (2009), 3-14.

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