We establish the uniqueness of the positive solution for equations of the form
−
Δ
u
=
a
u
−
b
(
x
)
f
(
u
)
-\Delta u=au-b(x)f(u)
in
Ω
\Omega
,
u
|
∂
Ω
=
∞
u|_{\partial \Omega }=\infty
. The special feature is to consider nonlinearities
f
f
whose variation at infinity is not regular (e.g.,
exp
(
u
)
−
1
\exp (u)-1
,
sinh
(
u
)
\sinh (u)
,
cosh
(
u
)
−
1
\cosh (u)-1
,
exp
(
u
)
log
(
u
+
1
)
\exp (u)\log (u+1)
,
u
β
exp
(
u
γ
)
u^\beta \exp (u^\gamma )
,
β
∈
R
\beta \in {\mathbb R}
,
γ
>
0
\gamma >0
or
exp
(
exp
(
u
)
)
−
e
\exp (\exp (u))-e
) and functions
b
≥
0
b\geq 0
in
Ω
\Omega
vanishing on
∂
Ω
\partial \Omega
. The main innovation consists of using Karamata’s theory not only in the statement/proof of the main result but also to link the nonregular variation of
f
f
at infinity with the blow-up rate of the solution near
∂
Ω
\partial \Omega
.