  We prove that flat ground state solutions ($i.e.$ minimizing the energy and
with gradient vanishing on the boundary of the domain) of the Dirichlet problem
associated to some semilinear autonomous elliptic equations with a strong
absorption term given by a non-Lipschitz function are unstable for dimensions
$N=1,2$ and they can be stable for $N\geq 3$ for suitable values of the
involved exponents.
