In this paper we consider singular semilinear elliptic equations whose prototype is the following −div A(x)Du = f(x)g(u) + l(x) inΩ, u = 0 on ∂Ω, where Ω is an open bounded set of R^N, N≥1, A is a bounded coercive matrix, g has a mild singularity at u=0, and f(x), l(x) are nonnegative functions in a convenient Lebesgue space . 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 Ω.

A semilinear elliptic equation with a mild singularities at u=0: Existence and homogeneization / Giachetti, Daniela; Aparicio, Pedro Martinez; Murat, Francois. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 0021-7824. - STAMPA. - 107 (1):(2017), pp. 41-77. [http://dx.doi.org/10.1016/j.matpur.2016.04.007]

A semilinear elliptic equation with a mild singularities at u=0: Existence and homogeneization

GIACHETTI, Daniela;
2017

Abstract

In this paper we consider singular semilinear elliptic equations whose prototype is the following −div A(x)Du = f(x)g(u) + l(x) inΩ, u = 0 on ∂Ω, where Ω is an open bounded set of R^N, N≥1, A is a bounded coercive matrix, g has a mild singularity at u=0, and f(x), l(x) are nonnegative functions in a convenient Lebesgue space . 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 Ω.
2017
Semilinear equations. Singularity at u =0. Existence. Stability. Uniqueness. Homogenization
01 Pubblicazione su rivista::01a Articolo in rivista
A semilinear elliptic equation with a mild singularities at u=0: Existence and homogeneization / Giachetti, Daniela; Aparicio, Pedro Martinez; Murat, Francois. - In: JOURNAL DE MATHÉMATIQUES PURES ET APPLIQUÉES. - ISSN 0021-7824. - STAMPA. - 107 (1):(2017), pp. 41-77. [http://dx.doi.org/10.1016/j.matpur.2016.04.007]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/868436
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