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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), 327-345. 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), 445-484. 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), 247-270. 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), 119-136 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), 159-183.  Google Scholar

[6]

C. O. Alves, J. V. Goncalves and L. Maia, Singular nonlinear elliptic equations in $\mathbbR^N$, Abstr. Appl. Anal., 3 (1998), 411-423. 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), 819-830. 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), 4006-4042. 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), 241-273.  Google Scholar

[10]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008), 411-426. doi: 10.1051/cocv:2008031.  Google Scholar

[11]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var., 37 (2010), 363-380. 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), 513-523. 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), 223-262.  Google Scholar

[14]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106. 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), 473-484. 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), 441-476. 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), 721-730. 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) 309-322.  Google Scholar

[20]

G. Stampacchia, Le problème de Dirichlet pour les équations élliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258. 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), 327-345. 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), 445-484. 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), 247-270. 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), 119-136 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), 159-183.  Google Scholar

[6]

C. O. Alves, J. V. Goncalves and L. Maia, Singular nonlinear elliptic equations in $\mathbbR^N$, Abstr. Appl. Anal., 3 (1998), 411-423. 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), 819-830. 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), 4006-4042. 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), 241-273.  Google Scholar

[10]

L. Boccardo, Dirichlet problems with singular and gradient quadratic lower order terms, ESAIM. Control, Optimisation and Calculus of Variations, 14 (2008), 411-426. doi: 10.1051/cocv:2008031.  Google Scholar

[11]

L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var., 37 (2010), 363-380. 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), 513-523. 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), 223-262.  Google Scholar

[14]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106. 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), 473-484. 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), 441-476. 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), 721-730. 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) 309-322.  Google Scholar

[20]

G. Stampacchia, Le problème de Dirichlet pour les équations élliptiques du second ordre à coefficients discontinus, Ann. Inst. Fourier, 15 (1965), 189-258. doi: 10.5802/aif.204.  Google Scholar

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