  We study the $\bar \partial $-equation first in Stein manifold then in
complete K\"ahler manifolds. The aim is to get $L^{r}$ and Sobolev estimates on
solutions with compact support.
  In the Stein case we get that for any $(p,q)$-form $\omega $ in $L^{r}$ with
compact support and $\bar \partial $-closed there is a $(p,q-1)$-form $u$ in
$W^{1,r}$ with compact support and such that $\bar \partial u=\omega .$
  In the case of K\"ahler manifold, we prove and use estimates on solutions on
Poisson equation with compact support and the link with $\bar \partial $
equation is done by a classical theorem stating that the Hodge laplacian is
twice the $\bar \partial $ (or Kohn) Laplacian in a K\"ahler manifold.
  This uses and improves, in special cases, our result on Andreotti-Grauert
type theorem.
