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

By Daniela Giachetti, Pedro J. Martínez-Aparicio and François Murat

Abstract

International audienceIn this paper we consider semilinear elliptic equations with mild singularities, whose prototype is the following ---------- −div A(x)Du = f (x)g(u) + l(x) in Ω, ---------- u = 0 on ∂Ω, ---------- where Ω is an open bounded set of R^N , N ≥ 1, A ∈ L^∞(Ω)^{N ×N} is a coercive matrix, g : [0, +∞) → [0, +∞] is continuous, and 0 ≤ g(s) ≤ 1 + 1/s^γ ∀ s > 0, with 0 < γ ≤ 1 and f, l ∈ L^r(Ω), r = 2N/(N +2) if N ≥ 3, r > 1 if N = 2, r = 1 if N = 1, f (x), l(x) ≥ 0 a.e. x ∈ Ω. ---------- We prove the existence of at least one nonnegative solution and a stability result; moreover uniqueness is also proved if g(s) is nonincreasing or "almost nonincreasing". Finally, we study the homogenization of these equations posed in a sequence of domains Ω ε obtained by removing many small holes from a fixed domain Ω

Topics: singularity, Semilinear equations, existence, uniqueness, stability, homogenization, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP]
Publisher: Elsevier
Year: 2015
DOI identifier: 10.1016/j.matpur.2016.04.007
OAI identifier: oai:HAL:hal-01157762v1
Provided by: Hal-Diderot

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