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

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.
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.   Google Scholar

[2]

H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271.   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. Poincar\'e Anal. Non Lin巃ire, 26 (2009), 1099.   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.   Google Scholar

[5]

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

[7]

J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,", The Clarendon Press, (1993).   Google Scholar

[8]

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

[9]

A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis,", Prentice-Hall, (1970).   Google Scholar

[10]

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

[12]

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

[13]

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

[14]

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

[16]

V. V. Kurta, On the absence of positive solutions of elliptic equations,, Mat. Zametki, 65 (1999), 552.   Google Scholar

[17]

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

[19]

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

[21]

J. Serrin, Entire solutions of quasilinear elliptic equations,, J. Math. Anal. Appl., 352 (2009), 3.   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, (2001), 35.   Google Scholar

[2]

H. Brezis, Semilinear equations in $R^N$ without condition at infinity,, Appl. Math. Optim., 12 (1984), 271.   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. Poincar\'e Anal. Non Lin巃ire, 26 (2009), 1099.   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.   Google Scholar

[5]

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

[7]

J. Heinonen, T. Kilpeläinen and O. Martio, "Nonlinear Potential Theory of Degenerate Elliptic Equations,", The Clarendon Press, (1993).   Google Scholar

[8]

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

[9]

A. N. Kolmogorov and S. V. Fomin, "Introductory Real Analysis,", Prentice-Hall, (1970).   Google Scholar

[10]

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

[12]

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

[13]

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

[14]

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

[16]

V. V. Kurta, On the absence of positive solutions of elliptic equations,, Mat. Zametki, 65 (1999), 552.   Google Scholar

[17]

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

[19]

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

[21]

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

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