We consider the boundary value problem
\[
−
Δ
u
(
x
)
=
λ
f
(
u
(
x
)
)
,
x
∈
Ω
B
u
(
x
)
=
0
,
x
∈
∂
Ω
\begin {gathered} - \Delta u(x) = \lambda f(u(x)),\quad x \in \Omega \hfill \\ Bu(x) = 0,\quad x \in \partial \Omega \hfill \\ \end {gathered}
\]
where
Ω
\Omega
is a bounded region in
R
N
{R^N}
with smooth boundary,
B
u
=
α
h
(
x
)
u
+
(
1
−
α
)
∂
u
/
∂
n
Bu = \alpha h(x)u + (1 - \alpha )\partial u/\partial n
where
α
∈
[
0
,
1
]
h
:
∂
Ω
→
R
+
\alpha \in [0,1]h:\partial \Omega \to {R^ + }
with
h
=
1
h = 1
when
α
=
1
\alpha = 1
,
λ
>
0
,
f
\lambda > 0,f
is a smooth function with
f
(
0
)
>
0
f(0) > 0
(semipositone),
f
′
(
u
)
>
0
f’(u) > 0
for
u
>
0
u > 0
and
f
(
u
)
≥
0
f(u) \geq 0
for
u
>
0
u > 0
. We prove that every nonnegative solution is unstable.