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

Quasilinear elliptic problem with Hardy potential and singular term

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.
Citation: Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262.

[14]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.

[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.

[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.

[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.

[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}, (). 

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

H. Brezis and X. Cabré, Some simple nonlinear PDE's without solutions, Boll. Unione. Mat. Ital. Sez. B, 8 (1998), 223-262.

[14]

H. Brezis and S. Kamin, Sublinear elliptic equations in $\mathbbR^N$, Manuscripta Math., 74 (1992), 87-106. doi: 10.1007/BF02567660.

[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.

[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.

[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.

[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}, (). 

[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.

[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.

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