# A semilinear elliptic problem with a singularity in $u = 0$

## François Murat

### (UPCM, Paris, France)

Date: Jul 04, 2014, time: 10:50

Abstract. In this joint work with Daniela Giachetti (Rome) and Pedro J. Martinez Aparicio (Cartagena, Spain) we consider the problem
$- div A(x) Du = F (x, u) \; {\rm in} \; \Omega,$
$u = 0 \; {\rm on} \; \partial \Omega,$
(namely an elliptic semilinear equation with homogeneous Dirichlet boundary condition),
where the non\-linearity $F (x, u)$ is singular in $u = 0$, with a singularity of the type
$F (x, u) = {f(x) \over u^\gamma} + g(x)$
with $\gamma > 0$ and $f$ and $g$ non negative (which implies that also $u$ is non negative).

The main difficulty is to give a convenient definition of the solution of the problem, in particular when $\gamma > 1$. We give such a definition and prove the existence and stability of a solution, as well as its uniqueness when $F(x, u)$ is non increasing en $u$.

We also consider the homogenization problem where $\Omega$ is replaced by $\Omega^\varepsilon$, with $\Omega^\varepsilon$ obtained from $\Omega$ by removing many very
small holes in such a way that passing to the limit when $\varepsilon$ tends to zero the Dirichlet boundary condition leads to an homogenized problem where a ''strange term" $\mu u$ appears.

This work was inspired by the paper of Lucio Boccardo and Luigi Orsina $Semilinear$ $elliptic$ $equations$ $with$ $singular$ $nonlinearities$, Calc. Var. Partial Differential Equations, 37, (2010), pp. 363--380.