American Institute of Mathematical Sciences

Advanced
May  2013, 12(3): 1363-1380. doi: 10.3934/cpaa.2013.12.1363

Quasilinear elliptic problem with Hardy potential and singular term

Boumediene Abdellaoui 1, and  Ahmed Attar 1,

1. 

Département de Mathématiques, Université Abou Bakr Belkaïd, Tlemcen, Tlemcen 13000, Algeria, Algeria

Received  April 2012 Revised  July 2012 Published  September 2012

We consider the following quasilinear elliptic problem \begin{eqnarray*} -\Delta_pu =\lambda\frac{u^{p-1}}{|x|^p}+\frac{h}{u^\gamma} \quad in \quad\Omega, \end{eqnarray*} where $1 < p < N, \Omega\subset R^N$ is a bounded regular domain such that $0\in \Omega, \gamma>0$ and $h$ is a nonnegative measurable function with suitable hypotheses.
The main goal of this work is to analyze the interaction between the Hardy potential and the singular term $u^{-\gamma}$ in order to get a solution for the largest possible class of the datum $h$. The regularity of the solution is also analyzed.
Keywords: comparison principle, singular Hardy-Sobolev potential, existence and nonexistence results., Quasilinear elliptic problems.
Mathematics Subject Classification: Primary: 35K15, 35K55, 35K65; Secondary: 35B05, 35B4.
Citation: 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
References:
[1]

B. Abdellaoui, E. Collorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities,, Calc. Var, 23 (2005), 327.  doi: 10.1007/s00526-004-0303-8.  Google Scholar

[2]

B. Abdellaoui, V. Felli and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-laplacian,, Boll. Unione Mat. Ital. Sez. B., 2 (2006), 445.  doi: 10.1007/s10231-002-0064-y.  Google Scholar

[3]

B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential,, Annal. Math. Pura. Appl, 182 (2003), 247.  doi: 10.1007/s10231-002-0064-y.  Google Scholar

[4]

B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient,, Jour. Euro. Math. Soc, 6 (2006), 119.  doi: 10.4171/JEMS/43.  Google Scholar

[5]

B. Abdellaoui and I. Peral, The Equation $-\Delta u-\lambda \fracu{|x|^2} = |\nabla u|^p +cf(x)$, the optimal power,, Ann. Scuola Norm. Sup. Pisa, 5 (2007), 159.   Google Scholar

[6]

C. O. Alves, J. V. Goncalves and L. Maia, Singular nonlinear elliptic equations in $\mathbbR^N$,, Abstr. Appl. Anal., 3 (1998), 411.  doi: 10.1155/S1085337598000633.  Google Scholar

[7]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications,, Nonlinear Ana. T.M.A., 32 (1998), 819.  doi: 10.1016/S0362-546X(97)00530-0.  Google Scholar

[8]

D. Arcoya, J. Carmona, T. Leonori, P. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations,, J. Differential Equations, 246 (2009), 4006.  doi: 10.1016/j.jde.2009.01.016.  Google Scholar

[9]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations,, Ann. Scuola Norm. Sup. Pisa. Cl. Sci., 22 (1995), 241.   Google Scholar

[10]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms,, ESAIM. Control, 14 (2008), 411.  doi: 10.1051/cocv:2008031.  Google Scholar

[11]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities,, Calc. Var., 37 (2010), 363.  doi: 10.1007/s00526-009-0266-x.  Google Scholar

[12]

L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential,, Discrete Contin. Dyn. Syst., 16 (2006), 513.  doi: 10.3934/dcds.2006.16.513.  Google Scholar

[13]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions,, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223.   Google Scholar

[14]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, Manuscripta Math., 74 (1992), 87.  doi: 10.1007/BF02567660.  Google Scholar

[15]

J. Cheng and Z. Zhang, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems,, Nonlinear Anal., 57 (2004), 473.  doi: 10.1016/j.na.2004.02.025.  Google Scholar

[16]

J. García Azorero and I. Peral, Hardy Inequalities and some critical elliptic and parabolic problems,, J. Diff. Eq., 144 (1998), 441.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

[17]

A. C. Lazer and J. P. McKenna, On a singular nonlinear elliptic boundary-value problem,, Proc. Amer. Math. Soc., 111 (1991), 721.  doi: 10.2307/2048410.  Google Scholar

[18]

S. E. Miri, Quasilinear elliptic problems with Hardy potential and reaction term,, Differ. Equ. Appl. Available from: \url{ http://dea.ele-math.com/forthcoming}, ().   Google Scholar

[19]

F. Murat, L'injection du cone positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$,, J. Math. Pures Appl., 60 (1981), 309.   Google Scholar

[20]

G. Stampacchia, Le problème de Dirichlet pour les équations élliptiques du second ordre à coefficients discontinus,, Ann. Inst. Fourier, 15 (1965), 189.  doi: 10.5802/aif.204.  Google Scholar

show all references

References:
[1]

B. Abdellaoui, E. Collorado and I. Peral, Some improved Caffarelli-Kohn-Nirenberg inequalities,, Calc. Var, 23 (2005), 327.  doi: 10.1007/s00526-004-0303-8.  Google Scholar

[2]

B. Abdellaoui, V. Felli and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-laplacian,, Boll. Unione Mat. Ital. Sez. B., 2 (2006), 445.  doi: 10.1007/s10231-002-0064-y.  Google Scholar

[3]

B. Abdellaoui and I. Peral, Existence and nonexistence results for quasilinear elliptic equations involving the p-Laplacian with a critical potential,, Annal. Math. Pura. Appl, 182 (2003), 247.  doi: 10.1007/s10231-002-0064-y.  Google Scholar

[4]

B. Abdellaoui and I. Peral, A note on a critical problem with natural growth in the gradient,, Jour. Euro. Math. Soc, 6 (2006), 119.  doi: 10.4171/JEMS/43.  Google Scholar

[5]

B. Abdellaoui and I. Peral, The Equation $-\Delta u-\lambda \fracu{|x|^2} = |\nabla u|^p +cf(x)$, the optimal power,, Ann. Scuola Norm. Sup. Pisa, 5 (2007), 159.   Google Scholar

[6]

C. O. Alves, J. V. Goncalves and L. Maia, Singular nonlinear elliptic equations in $\mathbbR^N$,, Abstr. Appl. Anal., 3 (1998), 411.  doi: 10.1155/S1085337598000633.  Google Scholar

[7]

W. Allegretto and Y. X. Huang, A Picone's identity for the $p$-Laplacian and applications,, Nonlinear Ana. T.M.A., 32 (1998), 819.  doi: 10.1016/S0362-546X(97)00530-0.  Google Scholar

[8]

D. Arcoya, J. Carmona, T. Leonori, P. Martínez-Aparicio, L. Orsina and F. Petitta, Existence and nonexistence of solutions for singular quadratic quasilinear equations,, J. Differential Equations, 246 (2009), 4006.  doi: 10.1016/j.jde.2009.01.016.  Google Scholar

[9]

P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J. L. Vazquez, An $L^1$-theory of existence and uniqueness of solutions of nonlinear elliptic equations,, Ann. Scuola Norm. Sup. Pisa. Cl. Sci., 22 (1995), 241.   Google Scholar

[10]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms,, ESAIM. Control, 14 (2008), 411.  doi: 10.1051/cocv:2008031.  Google Scholar

[11]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities,, Calc. Var., 37 (2010), 363.  doi: 10.1007/s00526-009-0266-x.  Google Scholar

[12]

L. Boccardo, L. Orsina and I. Peral, A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential,, Discrete Contin. Dyn. Syst., 16 (2006), 513.  doi: 10.3934/dcds.2006.16.513.  Google Scholar

[13]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions,, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223.   Google Scholar

[14]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$,, Manuscripta Math., 74 (1992), 87.  doi: 10.1007/BF02567660.  Google Scholar

[15]

J. Cheng and Z. Zhang, Existence and optimal estimates of solutions for singular nonlinear Dirichlet problems,, Nonlinear Anal., 57 (2004), 473.  doi: 10.1016/j.na.2004.02.025.  Google Scholar

[16]

J. García Azorero and I. Peral, Hardy Inequalities and some critical elliptic and parabolic problems,, J. Diff. Eq., 144 (1998), 441.  doi: 10.1006/jdeq.1997.3375.  Google Scholar

[17]

A. C. Lazer and J. P. McKenna, On a singular nonlinear elliptic boundary-value problem,, Proc. Amer. Math. Soc., 111 (1991), 721.  doi: 10.2307/2048410.  Google Scholar

[18]

S. E. Miri, Quasilinear elliptic problems with Hardy potential and reaction term,, Differ. Equ. Appl. Available from: \url{ http://dea.ele-math.com/forthcoming}, ().   Google Scholar

[19]

F. Murat, L'injection du cone positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$,, J. Math. Pures Appl., 60 (1981), 309.   Google Scholar

[20]

G. Stampacchia, Le problème de Dirichlet pour les équations élliptiques du second ordre à coefficients discontinus,, Ann. Inst. Fourier, 15 (1965), 189.  doi: 10.5802/aif.204.  Google Scholar

[1]

Xiaomei Sun, Wenyi Chen. Positive solutions for singular elliptic equations with critical Hardy-Sobolev exponent. Communications on Pure & Applied Analysis, 2011, 10 (2) : 527-540. doi: 10.3934/cpaa.2011.10.527
[2]

Jann-Long Chern, Yong-Li Tang, Chuan-Jen Chyan, Yi-Jung Chen. On the uniqueness of singular solutions for a Hardy-Sobolev equation. Conference Publications, 2013, 2013 (special) : 123-128. doi: 10.3934/proc.2013.2013.123
[3]

Jinhui Chen, Haitao Yang. A result on Hardy-Sobolev critical elliptic equations with boundary singularities. Communications on Pure & Applied Analysis, 2007, 6 (1) : 191-201. doi: 10.3934/cpaa.2007.6.191
[4]

Lucio Boccardo, Luigi Orsina, Ireneo Peral. A remark on existence and optimal summability of solutions of elliptic problems involving Hardy potential. Discrete & Continuous Dynamical Systems - A, 2006, 16 (3) : 513-523. doi: 10.3934/dcds.2006.16.513
[5]

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
[6]

Jing Zhang, Shiwang Ma. Positive solutions of perturbed elliptic problems involving Hardy potential and critical Sobolev exponent. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1999-2009. doi: 10.3934/dcdsb.2016033
[7]

Guoqing Zhang, Jia-yu Shao, Sanyang Liu. Linking solutions for N-laplace elliptic equations with Hardy-Sobolev operator and indefinite weights. Communications on Pure & Applied Analysis, 2011, 10 (2) : 571-581. doi: 10.3934/cpaa.2011.10.571
[8]

Boumediene Abdellaoui, Daniela Giachetti, Ireneo Peral, Magdalena Walias. Elliptic problems with nonlinear terms depending on the gradient and singular on the boundary: Interaction with a Hardy-Leray potential. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1747-1774. doi: 10.3934/dcds.2014.34.1747
[9]

Ilaria Fragalà, Filippo Gazzola, Gary Lieberman. Regularity and nonexistence results for anisotropic quasilinear elliptic equations in convex domains. Conference Publications, 2005, 2005 (Special) : 280-286. doi: 10.3934/proc.2005.2005.280
[10]

Dario D. Monticelli, Fabio Punzo. Nonexistence results for elliptic differential inequalities with a potential in bounded domains. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 675-695. doi: 10.3934/dcds.2018029
[11]

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
[12]

Masato Hashizume, Chun-Hsiung Hsia, Gyeongha Hwang. On the Neumann problem of Hardy-Sobolev critical equations with the multiple singularities. Communications on Pure & Applied Analysis, 2019, 18 (1) : 301-322. doi: 10.3934/cpaa.2019016
[13]

José Francisco de Oliveira, João Marcos do Ó, Pedro Ubilla. Hardy-Sobolev type inequality and supercritical extremal problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3345-3364. doi: 10.3934/dcds.2019138
[14]

Timothy Blass, Rafael De La Llave, Enrico Valdinoci. A comparison principle for a Sobolev gradient semi-flow. Communications on Pure & Applied Analysis, 2011, 10 (1) : 69-91. doi: 10.3934/cpaa.2011.10.69
[15]

Anna Maria Candela, Addolorata Salvatore. Existence of minimizers for some quasilinear elliptic problems. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020241
[16]

Fengshuang Gao, Yuxia Guo. Multiple solutions for a critical quasilinear equation with Hardy potential. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1977-2003. doi: 10.3934/dcdss.2019128
[17]

Evgeny Galakhov. Some nonexistence results for quasilinear PDE's. Communications on Pure & Applied Analysis, 2007, 6 (1) : 141-161. doi: 10.3934/cpaa.2007.6.141
[18]

Siegfried Carl. Comparison results for a class of quasilinear evolutionary hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 221-229. doi: 10.3934/proc.2007.2007.221
[19]

Wei Dai, Zhao Liu, Guozhen Lu. Hardy-Sobolev type integral systems with Dirichlet boundary conditions in a half space. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1253-1264. doi: 10.3934/cpaa.2017061
[20]

Yanfang Peng, Jing Yang. Sign-changing solutions to elliptic problems with two critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2015, 14 (2) : 439-455. doi: 10.3934/cpaa.2015.14.439

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences