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

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.