We prove uniqueness of positive radial solutions to the
p
p
-Laplacian problem
{
−
Δ
p
u
=
λ
f
(
u
)
in
Ω
,
u
=
0
on
∂
Ω
,
\begin{equation*} \left \{ \begin {array}{c} -\Delta _{p}u=\lambda f(u)\text { in }\Omega , \\ u=0\text { on }\partial \Omega , \end{array} \right . \end{equation*}
where
Δ
p
u
=
div
(
|
∇
u
|
p
−
2
∇
u
)
,
p
≥
2
,
Ω
\Delta _{p}u=\operatorname {div}(|\nabla u|^{p-2}\nabla u),p\geq 2,\ \Omega
is the open unit ball in
R
N
,
N
>
1
,
f
:
(
0
,
∞
)
→
R
R^{N}, N>1,\ f:(0,\infty )\rightarrow \mathbb {R}
is concave,
p
p
-sublinear at
∞
\infty
with infinite semipositone structure at
0
0
, and
λ
\lambda
is a large parameter.