A semilinear elliptic equation with a mild singularities at u=0: Existence and homogeneization
01 Pubblicazione su rivista
Giachetti Daniela, Aparicio Pedro Martinez, Murat Francois
ISSN: 0021-7824
In this paper we consider singular semilinear elliptic equations whose prototype is the following
?div A(x)Du = f(x)g(u) + l(x) in?,
u = 0 on ??,
where ? 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 ?.