In this paper, a semipositone anisotropic p-Laplacian problem $$ -\Delta _{\overrightarrow{p}}u=\lambda f(u), $$
−
Δ
p
→
u
=
λ
f
(
u
)
,
on a bounded domain with the Dirchlet boundary condition is considered, where $A(u^{q}-1)\leq f(u)\leq B(u^{q}-1)$
A
(
u
q
−
1
)
≤
f
(
u
)
≤
B
(
u
q
−
1
)
for $u>0$
u
>
0
, $f(0)<0$
f
(
0
)
<
0
and $f(u)=0$
f
(
u
)
=
0
for $u\leq -1$
u
≤
−
1
. It is proved that there exists $\lambda ^{*}>0$
λ
∗
>
0
such that if $\lambda \in (0,\lambda ^{*})$
λ
∈
(
0
,
λ
∗
)
, then the problem has a positive weak solution $u_{\lambda}\in L^{\infty}(\overline{\Omega})$
u
λ
∈
L
∞
(
Ω
‾
)
via combining Mountain-Pass arguments, comparison principles, and regularity principles.