Quasilinear elliptic problem with Hardy potential and singular term

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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

Abstract
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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

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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

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