# 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 & Continuous Dynamical Systems - A, 2017, 37 (1) : 15-31. doi: 10.3934/dcds.2017002
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