A semilinear elliptic equation with a mild singularity at u = 0: Existence and homogenization

Journal ar
Journal des Mathematiques Pures et Appliquees
  • Volumen: 107
  • Número: 1
  • Fecha: 01 enero 2017
  • Páginas: 41-77
  • ISSN: 00217824
  • Tipo de fuente: Revista
  • DOI: 10.1016/j.matpur.2016.04.007
  • Tipo de documento: Artículo
  • Editorial: Elsevier Masson SAS 62 rue Camille Desmoulins Issy les Moulineaux Cedex 92442
© 2016 Elsevier Masson SAS In this paper we consider singular semilinear elliptic equations whose prototype is the following {¿divA(x)Du=f(x)g(u)+l(x)in¿,u=0on¿¿, where ¿ is an open bounded set of RN,N¿1, A¿L¿(¿)N×N is a coercive matrix, g:[0,+¿[¿[0,+¿] is continuous, and 0¿g(s)¿1s¿+1 for every s>0, with 0<¿¿1 and f,l¿Lr(¿), r=2NN+2 if N¿3, r>1 if N=2, r=1 if N=1, f(x),l(x)¿0 a.e. x¿¿. We prove the existence of at least one nonnegative solution as well as a stability result; we also prove uniqueness if g(s) is nonincreasing or ¿almost nonincreasing¿. Finally, we study the homogenization of these equations posed in a sequence of domains ¿¿ obtained by removing many small holes from a fixed domain ¿.

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