American Institute of Mathematical Sciences

Advanced
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, (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

[1]

Genggeng Huang. A Liouville theorem of degenerate elliptic equation and its application. Discrete & Continuous Dynamical Systems - A, 2013, 33 (10) : 4549-4566. doi: 10.3934/dcds.2013.33.4549
[2]

Xiaohui Yu. Liouville type theorem for nonlinear elliptic equation with general nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4947-4966. doi: 10.3934/dcds.2014.34.4947
[3]

Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017
[4]

Thierry Horsin, Peter I. Kogut. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. I. Existence result. Mathematical Control & Related Fields, 2015, 5 (1) : 73-96. doi: 10.3934/mcrf.2015.5.73
[5]

Pengyan Wang, Pengcheng Niu. Liouville's theorem for a fractional elliptic system. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1545-1558. doi: 10.3934/dcds.2019067
[6]

Zongming Guo, Long Wei. A fourth order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2493-2508. doi: 10.3934/cpaa.2014.13.2493
[7]

Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
[8]

Fang-Fang Liao, Chun-Lei Tang. Four positive solutions of a quasilinear elliptic equation in $ R^N$. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2577-2600. doi: 10.3934/cpaa.2013.12.2577
[9]

Yinbin Deng, Wentao Huang. Positive ground state solutions for a quasilinear elliptic equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4213-4230. doi: 10.3934/dcds.2017179
[10]

Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361
[11]

Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363
[12]

Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198
[13]

Frank Arthur, Xiaodong Yan, Mingfeng Zhao. A Liouville-type theorem for higher order elliptic systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3317-3339. doi: 10.3934/dcds.2014.34.3317
[14]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949
[15]

Galina V. Grishina. On positive solution to a second order elliptic equation with a singular nonlinearity. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1335-1343. doi: 10.3934/cpaa.2010.9.1335
[16]

Zongming Guo, Yunting Yu. Boundary value problems for a semilinear elliptic equation with singular nonlinearity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 399-412. doi: 10.3934/cpaa.2016.15.399
[17]

Nicolas Forcadel, Mamdouh Zaydan. A comparison principle for Hamilton-Jacobi equation with moving in time boundary. Evolution Equations & Control Theory, 2019, 8 (3) : 543-565. doi: 10.3934/eect.2019026
[18]

Mehdi Badra, Kaushik Bal, Jacques Giacomoni. Existence results to a quasilinear and singular parabolic equation. Conference Publications, 2011, 2011 (Special) : 117-125. doi: 10.3934/proc.2011.2011.117
[19]

Ovidiu Savin. A Liouville theorem for solutions to the linearized Monge-Ampere equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 865-873. doi: 10.3934/dcds.2010.28.865
[20]

Frank Arthur, Xiaodong Yan. A Liouville-type theorem for higher order elliptic systems of Hé non-Lane-Emden type. Communications on Pure & Applied Analysis, 2016, 15 (3) : 807-830. doi: 10.3934/cpaa.2016.15.807

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (13)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences