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

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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

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 & 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, (2001), 35.
[2]	H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271.
[3]	L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type,, Ann. Inst. H. Poincar\'e Anal. Non Lin巃ire, 26 (2009), 1099.
[4]	L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities,, Nonlinear Anal., 70 (2009), 2882.
[5]	A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations,, J. Differ. Eqs., 250 (2011), 4367.
[6]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.
[7]	J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,", The Clarendon Press, (1993).
[8]	A. G. Kartsatos and R. D. Mabry, Controlling the space with preassigned responses,, J. Optim. Theory Appl., 54 (1987), 517.
[9]	A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis,", Prentice-Hall, (1970).
[10]	V. A. Kondrat$'$ev and E. M. Landis, Semilinear second-order equations with nonnegative characteristic form,, Mat. Zametki, 44 (1988), 457.
[11]	V. V. Kurta, Qualitative properties of solutions of some classes of second-order quasilinear elliptic equations,, Differentsial$'$nye Uravneniya, 28 (1992), 867.
[12]	V. V. Kurta, "Some Problems of Qualitative Theory for Nonlinear Second-order Equations,", Doctoral Dissert., (1994).
[13]	V. V. Kurta, On the comparison principle for second-order quasilinear elliptic equations,, Differentsial$'$nye Uravneniya, 31 (1995), 289.
[14]	V. V. Kurta, Comparison principle for solutions of parabolic inequalities,, C. R. Acad. Sci. Paris, 322 (1996), 1175.
[15]	V. V. Kurta, Comparison principle and analogues of the Phragmén-Lindelöf theorem for solutions of parabolic inequalities,, Appl. Anal., 71 (1999), 301.
[16]	V. V. Kurta, On the absence of positive solutions of elliptic equations,, Mat. Zametki, 65 (1999), 552.
[17]	J.-L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,", Dunod, (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.
[19]	E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities,, Dokl. Akad. Nauk, 359 (1998), 456.
[20]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79.
[21]	J. Serrin, Entire solutions of quasilinear elliptic equations,, J. Math. Anal. Appl., 352 (2009), 3.

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, (2001), 35.
[2]	H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271.
[3]	L. Damascelli, A. Farina, B. Sciunzi and E. Valdinoci, Liouville results for m-Laplace equations of Lane-Emden-Fowler type,, Ann. Inst. H. Poincar\'e Anal. Non Lin巃ire, 26 (2009), 1099.
[4]	L. Dupaigne and A. Farina, Liouville theorems for stable solutions of semilinear elliptic equations with convex nonlinearities,, Nonlinear Anal., 70 (2009), 2882.
[5]	A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations,, J. Differ. Eqs., 250 (2011), 4367.
[6]	B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations,, Comm. Pure Appl. Math., 34 (1981), 525.
[7]	J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,", The Clarendon Press, (1993).
[8]	A. G. Kartsatos and R. D. Mabry, Controlling the space with preassigned responses,, J. Optim. Theory Appl., 54 (1987), 517.
[9]	A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis,", Prentice-Hall, (1970).
[10]	V. A. Kondrat$'$ev and E. M. Landis, Semilinear second-order equations with nonnegative characteristic form,, Mat. Zametki, 44 (1988), 457.
[11]	V. V. Kurta, Qualitative properties of solutions of some classes of second-order quasilinear elliptic equations,, Differentsial$'$nye Uravneniya, 28 (1992), 867.
[12]	V. V. Kurta, "Some Problems of Qualitative Theory for Nonlinear Second-order Equations,", Doctoral Dissert., (1994).
[13]	V. V. Kurta, On the comparison principle for second-order quasilinear elliptic equations,, Differentsial$'$nye Uravneniya, 31 (1995), 289.
[14]	V. V. Kurta, Comparison principle for solutions of parabolic inequalities,, C. R. Acad. Sci. Paris, 322 (1996), 1175.
[15]	V. V. Kurta, Comparison principle and analogues of the Phragmén-Lindelöf theorem for solutions of parabolic inequalities,, Appl. Anal., 71 (1999), 301.
[16]	V. V. Kurta, On the absence of positive solutions of elliptic equations,, Mat. Zametki, 65 (1999), 552.
[17]	J.-L. Lions, "Quelques méthodes de résolution des problèmes aux limites non linéaires,", Dunod, (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.
[19]	E. Mitidieri and S. I. Pokhozhaev, Absence of global positive solutions of quasilinear elliptic inequalities,, Dokl. Akad. Nauk, 359 (1998), 456.
[20]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79.
[21]	J. Serrin, Entire solutions of quasilinear elliptic equations,, J. Math. Anal. Appl., 352 (2009), 3.

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