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

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• 作者：Daniela Giachetti^a ; ^{daniela.giachetti@sbai.uniroma1.it} ; Pedro J. Martí ; nez-Aparicio^b ; ^{pedroj.martinez@upct.es} ; Franç ; ois Murat^c ; ^{murat@ann.jussieu.fr}
• 刊名：Journal de Mathématiques Pures et Appliquées
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：107
• 期：1
• 页码：41-77
• 全文大小：651 K
• 卷排序：107

文摘

In this paper we consider singular semilinear elliptic equations whose prototype is the following{−divA(x)Du=f(x)g(u)+l(x)inΩ,u=0on∂Ω, where Ω is an open bounded set of RN,N≥1, A∈L∞(Ω)N×NA∈L∞(Ω)N×N is a coercive matrix, g:[0,+∞[→[0,+∞]g:[0,+∞[→[0,+∞] is continuous, and 0≤g(s)≤1sγ+1 for every s>0s>0, with 0<γ≤10<γ≤1 and f,l∈Lr(Ω)f,l∈Lr(Ω), r=2NN+2 if N≥3N≥3, r>1r>1 if N=2N=2, r=1r=1 if N=1N=1, f(x),l(x)≥0f(x),l(x)≥0 a.e. x∈Ωx∈Ω.We prove the existence of at least one nonnegative solution as well as a stability result; we also prove uniqueness if g(s)g(s) is nonincreasing or “almost nonincreasing”.Finally, we study the homogenization of these equations posed in a sequence of domains ΩεΩε obtained by removing many small holes from a fixed domain Ω.