American Institute of Mathematical Sciences

Advanced
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

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), 141-161. 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), 186-216.  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), 189-220. 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), 1-362.  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), 129-162. 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), 375-383. 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), 216-228.  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), 223-262.  Google Scholar

[9]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.  Google Scholar

[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.  Google Scholar

[11]

L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869. 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), 1-20. 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), 967-1020. 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), 1-49. doi: 10.1007/BF02788105.  Google Scholar

[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.  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), 155-179. 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), 3008-3019. 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), 689-697. 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), 501-535. 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), 273-284.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

E. Galakhov, Some nonexistence results for quasilinear PDE's, Commun. Pure Appl. Anal., 6 (2007), 141-161. 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), 186-216.  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), 189-220. 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), 1-362.  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), 129-162. 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), 375-383. 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), 216-228.  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), 223-262.  Google Scholar

[9]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142. doi: 10.1007/BF02392645.  Google Scholar

[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.  Google Scholar

[11]

L. D'Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal., 70 (2009), 2855-2869. 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), 1-20. 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), 967-1020. 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), 1-49. doi: 10.1007/BF02788105.  Google Scholar

[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.  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), 155-179. 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), 3008-3019. 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), 689-697. 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), 501-535. 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), 273-284.  Google Scholar

[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.  Google Scholar

[1]

Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. A new exact penalty function method for continuous inequality constrained optimization problems. Journal of Industrial & Management Optimization, 2010, 6 (4) : 895-910. doi: 10.3934/jimo.2010.6.895
[2]

Qinghua Ma, Zuoliang Xu, Liping Wang. Recovery of the local volatility function using regularization and a gradient projection method. Journal of Industrial & Management Optimization, 2015, 11 (2) : 421-437. doi: 10.3934/jimo.2015.11.421
[3]

Yanfei Wang, Qinghua Ma. A gradient method for regularizing retrieval of aerosol particle size distribution function. Journal of Industrial & Management Optimization, 2009, 5 (1) : 115-126. doi: 10.3934/jimo.2009.5.115
[4]

Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete & Continuous Dynamical Systems, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927
[5]

Yongjian Yang, Zhiyou Wu, Fusheng Bai. A filled function method for constrained nonlinear integer programming. Journal of Industrial & Management Optimization, 2008, 4 (2) : 353-362. doi: 10.3934/jimo.2008.4.353
[6]

Zhiyou Wu, Fusheng Bai, Guoquan Li, Yongjian Yang. A new auxiliary function method for systems of nonlinear equations. Journal of Industrial & Management Optimization, 2015, 11 (2) : 345-364. doi: 10.3934/jimo.2015.11.345
[7]

Liuyang Yuan, Zhongping Wan, Jingjing Zhang, Bin Sun. A filled function method for solving nonlinear complementarity problem. Journal of Industrial & Management Optimization, 2009, 5 (4) : 911-928. doi: 10.3934/jimo.2009.5.911
[8]

Li Wang, Yang Li, Liwei Zhang. A differential equation method for solving box constrained variational inequality problems. Journal of Industrial & Management Optimization, 2011, 7 (1) : 183-198. doi: 10.3934/jimo.2011.7.183
[9]

Suxiang He, Pan Zhang, Xiao Hu, Rong Hu. A sample average approximation method based on a D-gap function for stochastic variational inequality problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 977-987. doi: 10.3934/jimo.2014.10.977
[10]

Changjun Yu, Kok Lay Teo, Liansheng Zhang, Yanqin Bai. On a refinement of the convergence analysis for the new exact penalty function method for continuous inequality constrained optimization problem. Journal of Industrial & Management Optimization, 2012, 8 (2) : 485-491. doi: 10.3934/jimo.2012.8.485
[11]

E. B. Dynkin. A new inequality for superdiffusions and its applications to nonlinear differential equations. Electronic Research Announcements, 2004, 10: 68-77.

[12]

Saman Babaie–Kafaki, Reza Ghanbari. A class of descent four–term extension of the Dai–Liao conjugate gradient method based on the scaled memoryless BFGS update. Journal of Industrial & Management Optimization, 2017, 13 (2) : 649-658. doi: 10.3934/jimo.2016038
[13]

Yuri V. Rogovchenko, Fatoş Tuncay. Interval oscillation of a second order nonlinear differential equation with a damping term. Conference Publications, 2007, 2007 (Special) : 883-891. doi: 10.3934/proc.2007.2007.883
[14]

Alain Haraux. Some applications of the Łojasiewicz gradient inequality. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2417-2427. doi: 10.3934/cpaa.2012.11.2417
[15]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149
[16]

El-Sayed M.E. Mostafa. A nonlinear conjugate gradient method for a special class of matrix optimization problems. Journal of Industrial & Management Optimization, 2014, 10 (3) : 883-903. doi: 10.3934/jimo.2014.10.883
[17]

Gaohang Yu, Shanzhou Niu, Jianhua Ma. Multivariate spectral gradient projection method for nonlinear monotone equations with convex constraints. Journal of Industrial & Management Optimization, 2013, 9 (1) : 117-129. doi: 10.3934/jimo.2013.9.117
[18]

Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021032
[19]

Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096
[20]

Canghua Jiang, Zhiqiang Guo, Xin Li, Hai Wang, Ming Yu. An efficient adjoint computational method based on lifted IRK integrator and exact penalty function for optimal control problems involving continuous inequality constraints. Discrete & Continuous Dynamical Systems - S, 2020, 13 (6) : 1845-1865. doi: 10.3934/dcdss.2020109

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (72)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences