# American Institute of Mathematical Sciences

January  2017, 37(1): 15-31. doi: 10.3934/dcds.2017002

## Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes

 1 Departamento de Matemáticas, Universidad de Almería, Ctra. Sacramento s/n, La Cañada de San Urbano 04120, Almería, Spain 2 Departamento de Matemática Aplicada y Estadística, Campus Alfonso XIII, Universidad Politécnica de Cartagena, 30203, Murcia, Spain

* Corresponding author

Received  March 2016 Revised  August 2016 Published  November 2016

Fund Project: Research supported by MINECO-FEDER grant MTM2015-68210-P and Junta de Andalucía FQM-194 (first author) and FQM-116 (second author). Programa de Apoyo a la Investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia, reference 19461/PI/14 (second author).

In this paper we consider the homogenization problem for quasilinear elliptic equations with singularities in the gradient, whose model is the following
 $\begin{equation*}\begin{cases}\displaystyle -Δ u^\varepsilon + \frac{|\nabla u^\varepsilon|^2}{{(u^\varepsilon})^θ} = f (x)& \mbox{in} \; Ω^\varepsilon,\\u^\varepsilon = 0&\mbox{on} \; \partial Ω^\varepsilon,\\\end{cases}\end{equation*}$
where Ω is an open bounded set of
 $\mathbb{R}^N$
,
 $θ ∈ (0,1)$
and
 $f$
is positive function that belongs to a certain Lebesgue's space. The homogenization of these equations is posed in a sequence of domains
 $Ω^\varepsilon$
obtained by removing many small holes from a fixed domain Ω. We also give a corrector result.
Citation: José Carmona, Pedro J. Martínez-Aparicio. Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes. Discrete and Continuous Dynamical Systems, 2017, 37 (1) : 15-31. doi: 10.3934/dcds.2017002
##### References:
 [1] D. Arcoya, J. Carmona, T. Leonori, P. J. 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. [2] D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Bifurcation for Quasilinear Elliptic Singular BVP, Comm. Partial Differential Equations, 36 (2011), 670-692.  doi: 10.1080/03605302.2010.501835. [3] D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Comparison principle for elliptic equations in divergence with singular lower order terms having natural growth Commun. Contemp. Math. to appear. doi: 10.1142/S0219199716500139. [4] D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoam., 24 (2008), 597-616.  doi: 10.4171/RMI/548. [5] D. Arcoya and S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM Control Optim. Calc. Var., 16 (2010), 327-336.  doi: 10.1051/cocv:2008072. [6] L. Boccardo, Dirichlet problems with singular and quadratic gradient lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426.  doi: 10.1051/cocv:2008031. [7] L. Boccardo and J. Casado-Díaz, Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence, Asymptot. Anal., 86 (2014), 1-15. [8] J. Carmona, P. J. Martínez-Aparicio and A. Suárez, Existence and non-existence of positive solutions for nonlinear elliptic singular equations with natural growth, Nonlinear Anal., 89 (2013), 157-169.  doi: 10.1016/j.na.2013.05.015. [9] J. Casado-Díaz, Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains, J. Math. Pures Appl., 76 (1997), 431-476.  doi: 10.1016/S0021-7824(97)89958-8. [10] J. Casado-Díaz, Homogenization of a quasi-linear problem with quadratic growth in perforated domains: An example, Ann. Inst. H. Poincaré Anal. non linéaire, 14 (1997), 669-686.  doi: 10.1016/S0294-1449(97)80129-1. [11] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, Ⅰ et Ⅱ', In Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. Ⅱ and Vol. Ⅲ, ed. by H. Brezis and J. -L. Lions. Research Notes in Math. 60 and 70, Pitman, London, (1982), 98-138 and 154-178. English translation: D. Cioranescu and F. Murat, A strange term coming from nowhere. In Topics in mathematical modeling of composite materials, ed. by A. Cherkaev and R. V. Kohn. Progress in Nonlinear Differential Equations and their Applications 31, Birkhäuser, Boston, (1997), 44-93. [12] G. Dal Maso and A. Garroni, New results of the asymptotic behaviour of Dirichlet problems in perforated domains, Math. Models Methods Appl. Sci., 4 (1994), 373-407.  doi: 10.1142/S0218202594000224. [13] G. Dal Maso and F. Murat, Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 239-290. [14] G. Dal Maso and F. Murat, Asymptotic behavior and correctors for linear Dirichlet problems with simultaneously varying operators and domains, Ann. Inst. H. Poincaré Anal. non linéaire, 21 (2004), 445-486.  doi: 10.1016/j.anihpc.2003.05.001. [15] P. Donato and D. Giachetti, Homogenization of some nonlinear elliptic problems, Int. J. Pure Appl. Math., 73 (2011), 349-378. [16] D. Giachetti, P. J. Martínez-Aparicio and F. Murat, A semilinear elliptic equation with a mild singularity at u=0: Existence and homogenization J. Math. Pures Appl. to appear. doi: 10.1016/j.matpur.2016.04.007. [17] D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Definition, existence, stability and uniqueness of the solution to a semi-linear elliptic problem with a strong singularity at u = 0, preprint. [18] D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Homogenization of a Dirichlet semi-linear elliptic problem with a strong singularity at u = 0 in a domain with many small holes, preprint. [19] D. Giachetti and F. Murat, Elliptic problems with lower order terms having singular behaviour, Boll. Unione Mat. Ital., 2 (2009), 349-370. [20] J. Leray and J. L. Lions, Quelques résultats de Višik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. [21] V. A. Marčenko and E. Ya. Khruslov, Kraevye Zadachi v Oblastyakh s Melkozernistoi Granitsei, (Russian) Boundary value problems in domains with a fine-grained boundary, Izdat. "Naukova Dumka", Kiev, 1974. [22] G. Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, in Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que. , 1966.

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##### References:
 [1] D. Arcoya, J. Carmona, T. Leonori, P. J. 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. [2] D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Bifurcation for Quasilinear Elliptic Singular BVP, Comm. Partial Differential Equations, 36 (2011), 670-692.  doi: 10.1080/03605302.2010.501835. [3] D. Arcoya, J. Carmona and P. J. Martínez-Aparicio, Comparison principle for elliptic equations in divergence with singular lower order terms having natural growth Commun. Contemp. Math. to appear. doi: 10.1142/S0219199716500139. [4] D. Arcoya and P. J. Martínez-Aparicio, Quasilinear equations with natural growth, Rev. Mat. Iberoam., 24 (2008), 597-616.  doi: 10.4171/RMI/548. [5] D. Arcoya and S. Segura de León, Uniqueness of solutions for some elliptic equations with a quadratic gradient term, ESAIM Control Optim. Calc. Var., 16 (2010), 327-336.  doi: 10.1051/cocv:2008072. [6] L. Boccardo, Dirichlet problems with singular and quadratic gradient lower order terms, ESAIM Control Optim. Calc. Var., 14 (2008), 411-426.  doi: 10.1051/cocv:2008031. [7] L. Boccardo and J. Casado-Díaz, Some properties of solutions of some semilinear elliptic singular problems and applications to the G-convergence, Asymptot. Anal., 86 (2014), 1-15. [8] J. Carmona, P. J. Martínez-Aparicio and A. Suárez, Existence and non-existence of positive solutions for nonlinear elliptic singular equations with natural growth, Nonlinear Anal., 89 (2013), 157-169.  doi: 10.1016/j.na.2013.05.015. [9] J. Casado-Díaz, Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains, J. Math. Pures Appl., 76 (1997), 431-476.  doi: 10.1016/S0021-7824(97)89958-8. [10] J. Casado-Díaz, Homogenization of a quasi-linear problem with quadratic growth in perforated domains: An example, Ann. Inst. H. Poincaré Anal. non linéaire, 14 (1997), 669-686.  doi: 10.1016/S0294-1449(97)80129-1. [11] D. Cioranescu and F. Murat, Un terme étrange venu d'ailleurs, Ⅰ et Ⅱ', In Nonlinear partial differential equations and their applications, Collège de France Seminar, Vol. Ⅱ and Vol. Ⅲ, ed. by H. Brezis and J. -L. Lions. Research Notes in Math. 60 and 70, Pitman, London, (1982), 98-138 and 154-178. English translation: D. Cioranescu and F. Murat, A strange term coming from nowhere. In Topics in mathematical modeling of composite materials, ed. by A. Cherkaev and R. V. Kohn. Progress in Nonlinear Differential Equations and their Applications 31, Birkhäuser, Boston, (1997), 44-93. [12] G. Dal Maso and A. Garroni, New results of the asymptotic behaviour of Dirichlet problems in perforated domains, Math. Models Methods Appl. Sci., 4 (1994), 373-407.  doi: 10.1142/S0218202594000224. [13] G. Dal Maso and F. Murat, Asymptotic behavior and correctors for Dirichlet problems in perforated domains with homogeneous monotone operators, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 239-290. [14] G. Dal Maso and F. Murat, Asymptotic behavior and correctors for linear Dirichlet problems with simultaneously varying operators and domains, Ann. Inst. H. Poincaré Anal. non linéaire, 21 (2004), 445-486.  doi: 10.1016/j.anihpc.2003.05.001. [15] P. Donato and D. Giachetti, Homogenization of some nonlinear elliptic problems, Int. J. Pure Appl. Math., 73 (2011), 349-378. [16] D. Giachetti, P. J. Martínez-Aparicio and F. Murat, A semilinear elliptic equation with a mild singularity at u=0: Existence and homogenization J. Math. Pures Appl. to appear. doi: 10.1016/j.matpur.2016.04.007. [17] D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Definition, existence, stability and uniqueness of the solution to a semi-linear elliptic problem with a strong singularity at u = 0, preprint. [18] D. Giachetti, P. J. Martínez-Aparicio and F. Murat, Homogenization of a Dirichlet semi-linear elliptic problem with a strong singularity at u = 0 in a domain with many small holes, preprint. [19] D. Giachetti and F. Murat, Elliptic problems with lower order terms having singular behaviour, Boll. Unione Mat. Ital., 2 (2009), 349-370. [20] J. Leray and J. L. Lions, Quelques résultats de Višik sur les problémes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), 97-107. [21] V. A. Marčenko and E. Ya. Khruslov, Kraevye Zadachi v Oblastyakh s Melkozernistoi Granitsei, (Russian) Boundary value problems in domains with a fine-grained boundary, Izdat. "Naukova Dumka", Kiev, 1974. [22] G. Stampacchia, Èquations elliptiques du second ordre à coefficients discontinus, in Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965) Les Presses de l'Université de Montréal, Montreal, Que. , 1966.
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