The Dirichlet problem
Δ
u
=
λ
f
(
u
)
\Delta u = \lambda \,f(u)
in a domain
Ω
,
u
=
1
\Omega ,\,u = 1
on
∂
Ω
\partial \Omega
is considered with
f
(
t
)
=
0
f(t) = 0
if
t
≤
0
,
f
(
t
)
>
0
t \leq 0,\,f(t) > 0
if
t
>
0
,
f
(
t
)
∼
t
p
t > 0,\,f(t) \sim {t^p}
if
t
↓
0
,
0
>
p
>
1
;
f
(
t
)
t \downarrow 0,0 > p > 1;f(t)
is not monotone in general. The set
{
u
=
0
}
\{ u = 0\}
and the “free boundary”
∂
{
u
=
0
}
\partial \{ u = 0\}
are studied. Sharp asymptotic estimates are established as
λ
→
∞
\lambda \to \infty
. For suitable
f
f
, under the assumption that
Ω
\Omega
is a two-dimensional convex domain, it is shown that
{
u
=
0
}
\{ u = 0\}
is a convex set. Analogous results are established also in the case where
∂
u
/
∂
v
+
μ
(
u
−
1
)
=
0
\partial u/\partial v + \mu (u - 1) = 0
on
∂
Ω
\partial \Omega
.