  We consider equation $-\Delta u+f(x,u)=0$ in smooth bounded domain
$\Omega\in\mathbb{R}^N$, $N\geqslant2$, with $f(x,r)>0$ in
$\Omega\times\mathbb{R}^1_+$ and $f(x,r)=0$ on $\partial\Omega$. We find the
condition on the order of degeneracy of $f(x,r)$ near $\partial\Omega$, which
is a criterion of the existence-nonexistence of a very singular solution with a
strong point singularity on $\partial\Omega$. Moreover, we prove that the
mentioned condition is a sufficient condition for the uniqueness of a large
solution and conjecture that this condition is also a necessary condition of
the uniqueness.
