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Singular quasilinear elliptic problems with changing sign datum: existence and homogenization. (English) Zbl 1437.35036

The authors study singular quasilinear elliptic problems with changing sign datum. First the authors consider, under suitable assumptions, the boundary value problem \[ \left\{ \begin{array}{ll} -\mathrm{div}(M(x)\nabla u)=\lambda u+g(x,u)|\nabla u|^q+f(x)& \mathrm{in}\ \Omega\, ,\\ u=0& \mathrm{on}\ \partial \Omega\, , \end{array} \right. \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\). In the problem above, in particular, the function \(f\) belongs to \(L^p(\Omega)\) with \(p>N/2\) and no assumption on its sign is imposed. The first result concerns the existence of a solution for the quasilinear problem under investigation. Then the authors consider the homogenization problem \[ \left\{ \begin{array}{ll} -\mathrm{div}(M(x)\nabla u^\epsilon)=\lambda u^\epsilon+g(x,u^\epsilon)|\nabla u^\epsilon|^q+f(x)& \mathrm{in}\ \Omega^\epsilon\, ,\\ u^\epsilon=0& \mathrm{on}\ \partial \Omega^\epsilon\, , \end{array} \right. \] where \(\Omega^\epsilon\) is a domain obtained by removing many small holes from \(\Omega\). As second result of the paper, the authors investigate the properties of the strange term which appears in the limit problem as \(\epsilon \to 0\).

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35J25 Boundary value problems for second-order elliptic equations
35J62 Quasilinear elliptic equations
35J75 Singular elliptic equations
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[1] Arcoya, D.; Boccardo, L.; Leonori, T.; Porretta, A., Some elliptic problems with singular natural growth lower order terms, J. Differ. Equ., 249, 2771-2795 (2010) · Zbl 1203.35103 · doi:10.1016/j.jde.2010.05.009
[2] Arcoya, D.; De Coster, C.; Jeanjean, L.; Tanaka, K., Remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions, J. Math. Anal. Appl., 420, 772-780 (2014) · Zbl 1296.35068 · doi:10.1016/j.jmaa.2014.06.007
[3] Arcoya, D.; Moreno-Mérida, L., The effect of a singular term in a quadratic quasi-linear problem, J. Fixed Point Theory Appl., 19, 815-831 (2017) · Zbl 1376.35075 · doi:10.1007/s11784-016-0374-0
[4] Boccardo, L.; Murat, F.; Puel, J-P, Quelques propriétés des opérateurs elliptiques quasi linéaires, C. R. Acad. Sci. Paris Sér. I Math., 307, 749-752 (1988) · Zbl 0696.35050
[5] Carmona, J.; Leonori, T.; López-Martínez, S.; Martínez-Aparicio, Pj, Quasilinear elliptic problems with singular and homogeneous lower order terms, Nonlinear Anal., 179, 105-130 (2019) · Zbl 1404.35309 · doi:10.1016/j.na.2018.08.002
[6] Carmona, J.; Martínez-Aparicio, Pj, Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes, Discrete Contin. Dyn. Syst., 37, 1, 15-31 (2017) · Zbl 1357.35026 · doi:10.3934/dcds.2017002
[7] Casado-Díaz, J., Homogenization of general quasi-linear Dirichlet problems with quadratic growth in perforated domains, J. Math. Pures Appl., 76, 431-476 (1997) · Zbl 0880.35015 · doi:10.1016/S0021-7824(97)89958-8
[8] Casado-Díaz, J., Homogenization of a quasi-linear problem with quadratic growth in perforated domains: an example, Ann. l’Inst. Henri Poincare (C) Nonlinear Anal., 14, 669-686 (1997) · Zbl 0942.35051 · doi:10.1016/S0294-1449(97)80129-1
[9] Casado-Díaz, J., Homogenization of Dirichlet pseudomonotone problems with renormalized solutions in perforated domains, J. Math. Pures Appl. (9), 79, 6, 553-590 (2000) · Zbl 0957.35016 · doi:10.1016/S0021-7824(00)00151-3
[10] Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs, I et II. In: Brezis, H., Lions, J.-L. (eds.) Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar. Research Notes in Math. 60 and 70, Vols. II-III. Pitman, London (1982), 98-138 and 154-178. English translation: Cioranescu, D., Murat, F.: A strange term coming from nowhere. In: Cherkaev, A., Kohn, R.V. (eds.) Topics in Mathematical Modeling of Composite Materials. Progress in Nonlinear Differential Equations and their Applications, vol. 31, pp. 44-93. Birkhäuger, Boston (1997) · Zbl 0498.35034
[11] Dal Maso, G.; Garroni, A., New results of the asymptotic behaviour of Dirichlet problems in perforated domains, Math. Models Methods Appl. Sci., 3, 373-407 (1994) · Zbl 0804.47050 · doi:10.1142/S0218202594000224
[12] Giachetti, D.; Martínez-Aparicio, Pj; Murat, F., A semilinear elliptic equation with a mild singularity at \(\text{u} = 0\): existence and homogenization, J. Math. Pures Appl., 107, 41-77 (2017) · Zbl 1371.35112 · doi:10.1016/j.matpur.2016.04.007
[13] Giachetti, D.; Martínez-Aparicio, Pj; Murat, F., On the definition of the solution to a semilinear elliptic problem with a strong singularity at \(\text{ u } = 0\), Nonlinear Anal., 177, 491-523 (2018) · Zbl 1408.35026 · doi:10.1016/j.na.2018.04.023
[14] Giachetti, D.; Petitta, F.; Segura De León, S., Elliptic equations having a singular quadratic gradient term and a changing sign datum, Commun. Pure Appl. Anal., 11, 1875-1895 (2012) · Zbl 1266.35103 · doi:10.3934/cpaa.2012.11.1875
[15] Giachetti, D.; Petitta, F.; Segura De León, S., A priori estimates for elliptic problems with a strongly singular gradient term and a general datum, Differ. Integral Equ., 26, 9-10, 913-948 (2013) · Zbl 1299.35139
[16] Giachetti, D.; Segura De León, S., Quasilinear stationary problems with a quadratic gradient term having singularities, J. Lond. Math. Soc. (2), 86, 2, 585-606 (2012) · Zbl 1253.35064 · doi:10.1112/jlms/jds014
[17] Ladyzenskaya, O., Ural’tseva, N.: Linear and quasilinear elliptic equations. Translated from the Russian by Scripta Technica. Academic, New York (1968), xviii+495 pp (1968) · Zbl 0164.13002
[18] Leray, J.; Lions, Jl, Quelques résultats de Višik sur les problèmes elliptiques non linéaires par les méthodes de Minty-Browder, Bull. Soc. Math. Fr., 93, 97-107 (1965) · Zbl 0132.10502 · doi:10.24033/bsmf.1617
[19] Marcenko, V.A., Hruslov, E.J.: Boundary Value Problems in Domains with Fine-Grained Boundary. Naukova Dumka, Kiev (1974) (in Russian) · Zbl 0289.35002
[20] Stampacchia, G., Equations Èlliptiques du second ordre à coefficients discontinus, Les Presses de l’Université de Montréal Montreal Que, 35, 45, 326 (1966) · Zbl 0151.15501
[21] Troianiello, Giovanni Maria, Nonvariational Obstacle Problems, Elliptic Differential Equations and Obstacle Problems, 291-334 (1987), Boston, MA: Springer US, Boston, MA · Zbl 0655.35002
[22] Trudinger, Ns, Linear elliptic operators with measurable coefficients, Ann. Scuola Norm. Sup. Pisa (3), 27, 265-308 (1973) · Zbl 0279.35025
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