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

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 & 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. doi: 10.1002/cpa.3160390306. Google Scholar
[2]	E. Mitidieri and S. I. Pohozaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbbR^N$,, Proc. Steklov Inst. Math., 227 (1999), 186. Google Scholar
[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. doi: 10.1007/PL00001368. Google Scholar
[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. Google Scholar
[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. doi: 10.1007/s00032-004-0032-7. Google Scholar
[6]	G. Karisti, On the absence of solutions of systems of quasilinear elliptic inequalities,, Differ. Equs., 38 (2002), 375. doi: 10.1023/A:1016061909813. Google Scholar
[7]	G. G. Laptev, On the absence of solutions to a class of singular semilinear differential inequalities,, Proc. Steklov Inst. Math., 232 (2001), 216. Google Scholar
[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. Google Scholar
[9]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79. doi: 10.1007/BF02392645. Google Scholar
[10]	J. Serrin, Entire solutions of quasilinear elliptic equations,, J. Math. Anal. Appl., 352 (2009), 3. doi: 10.1016/j.jmaa.2008.10.036. Google Scholar
[11]	L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities,, Nonlinear Anal., 70 (2009), 2855. doi: 10.1016/j.na.2008.12.028. Google Scholar
[12]	L. D'Ambrosio and E. Mitidieri, Nonnegative solutions of some quasilinear elliptic inequalities and applications,, Mat. Sb., 201 (2010), 1. doi: 10.1070/SM2010v201n06ABEH004094. Google Scholar
[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. doi: 10.1016/j.aim.2009.12.017. Google Scholar
[14]	M. F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems,, J. Anal. Math., 84 (2001), 1. doi: 10.1007/BF02788105. Google Scholar
[15]	R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities,, Nonlinear Anal., 70 (2009), 2903. doi: 10.1016/j.na.2008.12.018. Google Scholar
[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. doi: 10.1017/s00205-007-0081-5. Google Scholar
[17]	R. Filippucci, P. Pucci and M. Rigoli, On weak solutions of nonlinear weighted $p$-Laplacian elliptic inequalities,, Nonlinear Anal., 70 (2009), 3008. doi: 10.1016/j.na.2008.12.031. Google Scholar
[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. doi: 10.1016/j.jmaa.2009.03.050. Google Scholar
[19]	R. Filippucci, P. Pucci and M. Rigoli, Nonlinear weighted $p$-Laplacian elliptic inequalities with gradient terms,, Commun. Contemp. Math., 12 (2010), 501. doi: 10.1142/S0219199710003841. Google Scholar
[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. Google Scholar
[21]	S. I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 487. doi: 10.1016/j.anihpc.2003.06.002. Google Scholar

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References:

[1]	E. Galakhov, Some nonexistence results for quasilinear PDE's,, Commun. Pure Appl. Anal., 6 (2007), 141. doi: 10.1002/cpa.3160390306. Google Scholar
[2]	E. Mitidieri and S. I. Pohozaev, Absence of positive solutions for quasilinear elliptic problems in $\mathbbR^N$,, Proc. Steklov Inst. Math., 227 (1999), 186. Google Scholar
[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. doi: 10.1007/PL00001368. Google Scholar
[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. Google Scholar
[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. doi: 10.1007/s00032-004-0032-7. Google Scholar
[6]	G. Karisti, On the absence of solutions of systems of quasilinear elliptic inequalities,, Differ. Equs., 38 (2002), 375. doi: 10.1023/A:1016061909813. Google Scholar
[7]	G. G. Laptev, On the absence of solutions to a class of singular semilinear differential inequalities,, Proc. Steklov Inst. Math., 232 (2001), 216. Google Scholar
[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. Google Scholar
[9]	J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities,, Acta Math., 189 (2002), 79. doi: 10.1007/BF02392645. Google Scholar
[10]	J. Serrin, Entire solutions of quasilinear elliptic equations,, J. Math. Anal. Appl., 352 (2009), 3. doi: 10.1016/j.jmaa.2008.10.036. Google Scholar
[11]	L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities,, Nonlinear Anal., 70 (2009), 2855. doi: 10.1016/j.na.2008.12.028. Google Scholar
[12]	L. D'Ambrosio and E. Mitidieri, Nonnegative solutions of some quasilinear elliptic inequalities and applications,, Mat. Sb., 201 (2010), 1. doi: 10.1070/SM2010v201n06ABEH004094. Google Scholar
[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. doi: 10.1016/j.aim.2009.12.017. Google Scholar
[14]	M. F. Bidaut-Véron and S. Pohozaev, Nonexistence results and estimates for some nonlinear elliptic problems,, J. Anal. Math., 84 (2001), 1. doi: 10.1007/BF02788105. Google Scholar
[15]	R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities,, Nonlinear Anal., 70 (2009), 2903. doi: 10.1016/j.na.2008.12.018. Google Scholar
[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. doi: 10.1017/s00205-007-0081-5. Google Scholar
[17]	R. Filippucci, P. Pucci and M. Rigoli, On weak solutions of nonlinear weighted $p$-Laplacian elliptic inequalities,, Nonlinear Anal., 70 (2009), 3008. doi: 10.1016/j.na.2008.12.031. Google Scholar
[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. doi: 10.1016/j.jmaa.2009.03.050. Google Scholar
[19]	R. Filippucci, P. Pucci and M. Rigoli, Nonlinear weighted $p$-Laplacian elliptic inequalities with gradient terms,, Commun. Contemp. Math., 12 (2010), 501. doi: 10.1142/S0219199710003841. Google Scholar
[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. Google Scholar
[21]	S. I. Pohozaev and A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities,, Ann. Inst. H. Poincar$\acutee$ Anal. Non Lin$\acutee$aire, 21 (2004), 487. doi: 10.1016/j.anihpc.2003.06.002. Google Scholar

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