Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities

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May 2012, 11(3): 935-943. doi: 10.3934/cpaa.2012.11.935

Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities

Xiaohong Li ^1, and Fengquan Li ^1,

1.	School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China, China

Received August 2010 Revised October 2011 Published December 2011

Abstract
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The aim of this paper is to prove some nonexistence results for nonnegative weak solutions of the nonlinear differential inequalities with gradient nonlinearities in $R^N$. The proofs are based on the test function method developed by Bidaut-Véron, Mitidieri and Pohozaev in [3] and [14].

Keywords: gradient term., Nonlinear differential inequality, test function method.

Mathematics Subject Classification: Primary: 35J60, 35J70; Secondary: 35R4.

Citation: Xiaohong Li, Fengquan Li. Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities. Communications on Pure and Applied Analysis, 2012, 11 (3) : 935-943. doi: 10.3934/cpaa.2012.11.935

References:

[1]	E. Galakhov, Some nonexistence results for quasilinear PDE's, Commun. Pure Appl. Anal., 6 (2007), 141-161. doi: 10.1002/cpa.3160390306.
[2]	E. Mitidieri and S. I. Pohozaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbb{R}^N2$, Proc. Steklov Inst. Math., 227 (1999), 186-216.
[3]	E. Mitidieri and S. I. Pohozaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $R^n$, J. Evol. Equ., 1 (2001), 189-220. doi: 10.1007/PL00001368.
[4]	E. Mitidieri and S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
[5]	E. Mitidieri and S. I. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129-162. doi: 10.1007/s00032-004-0032-7.
[6]	G. Karisti, On the absence of solutions of systems of quasilinear elliptic inequalities, Differ. Equs., 38 (2002), 375-383. doi: 10.1023/A:1016061909813.
[7]	G. G. Laptev, On the absence of solutions to a class of singular semilinear differential inequalities, Proc. Steklov Inst. Math., 232 (2001), 216-228.
[8]	H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 223-262.
[9]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.
[10]	J. Serrin, Entire solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 352 (2009), 3-14. doi: 10.1016/j.jmaa.2008.10.036.
[11]	L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869. doi: 10.1016/j.na.2008.12.028.
[12]	L. D'Ambrosio and E. Mitidieri, Nonnegative solutions of some quasilinear elliptic inequalities and applications, Mat. Sb., 201 (2010), 1-20. doi: 10.1070/SM2010v201n06ABEH004094.
[13]	L. D'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math., 224 (2010), 967-1020. doi: 10.1016/j.aim.2009.12.017.
[14]	M. F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49. doi: 10.1007/BF02788105.
[15]	R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916. doi: 10.1016/j.na.2008.12.018.
[16]	R. Filippucci, P. Pucci and M. Rigoli, Non-existence of entire solutions of degenerate elliptic inequalities with weights, Arch. Ration. Mech. Anal., 188 (2008), 155-179. doi: 10.1017/s00205-007-0081-5.
[17]	R. Filippucci, P. Pucci and M. Rigoli, On weak solutions of nonlinear weighted $p$-Laplacian elliptic inequalities, Nonlinear Anal., 70 (2009), 3008-3019. doi: 10.1016/j.na.2008.12.031.
[18]	R. Filippucci, P. Pucci and M. Rigoli, On entire solutions of degenerate elliptic differential inequalities with nonlinear gradient terms, J. Math. Anal. Appl., 356 (2009), 689-697. doi: 10.1016/j.jmaa.2009.03.050.
[19]	R. Filippucci, P. Pucci and M. Rigoli, Nonlinear weighted $p$-Laplacian elliptic inequalities with gradient terms, Commun. Contemp. Math., 12 (2010), 501-535. doi: 10.1142/S0219199710003841.
[20]	S. I. Pohozaev, A general approach to the theory of the nonexistence of global solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 236 (2002), 273-284.
[21]	S. I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 487-502. doi: 10.1016/j.anihpc.2003.06.002.

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References:

[1]	E. Galakhov, Some nonexistence results for quasilinear PDE's, Commun. Pure Appl. Anal., 6 (2007), 141-161. doi: 10.1002/cpa.3160390306.
[2]	E. Mitidieri and S. I. Pohozaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbb{R}^N2$, Proc. Steklov Inst. Math., 227 (1999), 186-216.
[3]	E. Mitidieri and S. I. Pohozaev, Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on $R^n$, J. Evol. Equ., 1 (2001), 189-220. doi: 10.1007/PL00001368.
[4]	E. Mitidieri and S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1-362.
[5]	E. Mitidieri and S. I. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129-162. doi: 10.1007/s00032-004-0032-7.
[6]	G. Karisti, On the absence of solutions of systems of quasilinear elliptic inequalities, Differ. Equs., 38 (2002), 375-383. doi: 10.1023/A:1016061909813.
[7]	G. G. Laptev, On the absence of solutions to a class of singular semilinear differential inequalities, Proc. Steklov Inst. Math., 232 (2001), 216-228.
[8]	H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., 1 (1998), 223-262.
[9]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.
[10]	J. Serrin, Entire solutions of quasilinear elliptic equations, J. Math. Anal. Appl., 352 (2009), 3-14. doi: 10.1016/j.jmaa.2008.10.036.
[11]	L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869. doi: 10.1016/j.na.2008.12.028.
[12]	L. D'Ambrosio and E. Mitidieri, Nonnegative solutions of some quasilinear elliptic inequalities and applications, Mat. Sb., 201 (2010), 1-20. doi: 10.1070/SM2010v201n06ABEH004094.
[13]	L. D'Ambrosio and E. Mitidieri, A priori estimates, positivity results, and nonexistence theorems for quasilinear degenerate elliptic inequalities, Adv. Math., 224 (2010), 967-1020. doi: 10.1016/j.aim.2009.12.017.
[14]	M. F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems, J. Anal. Math., 84 (2001), 1-49. doi: 10.1007/BF02788105.
[15]	R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916. doi: 10.1016/j.na.2008.12.018.
[16]	R. Filippucci, P. Pucci and M. Rigoli, Non-existence of entire solutions of degenerate elliptic inequalities with weights, Arch. Ration. Mech. Anal., 188 (2008), 155-179. doi: 10.1017/s00205-007-0081-5.
[17]	R. Filippucci, P. Pucci and M. Rigoli, On weak solutions of nonlinear weighted $p$-Laplacian elliptic inequalities, Nonlinear Anal., 70 (2009), 3008-3019. doi: 10.1016/j.na.2008.12.031.
[18]	R. Filippucci, P. Pucci and M. Rigoli, On entire solutions of degenerate elliptic differential inequalities with nonlinear gradient terms, J. Math. Anal. Appl., 356 (2009), 689-697. doi: 10.1016/j.jmaa.2009.03.050.
[19]	R. Filippucci, P. Pucci and M. Rigoli, Nonlinear weighted $p$-Laplacian elliptic inequalities with gradient terms, Commun. Contemp. Math., 12 (2010), 501-535. doi: 10.1142/S0219199710003841.
[20]	S. I. Pohozaev, A general approach to the theory of the nonexistence of global solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 236 (2002), 273-284.
[21]	S. I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 487-502. doi: 10.1016/j.anihpc.2003.06.002.

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