We consider the following Gelfand problem
(
P
)
λ
{
−
Δ
u
=
λ
a
(
x
)
f
(
u
)
a
m
p
;
in
Ω
,
u
>
0
a
m
p
;
in
Ω
,
u
=
0
a
m
p
;
on
∂
Ω
,
\begin{equation*} (P)_\lambda \qquad \left \{\begin {array}{ll} -\Delta u = \lambda a(x) f(u) & \text { in } \Omega , \\ u>0 & \text { in } \Omega , \\ u= 0 & \text { on } \partial \Omega , \end{array}\right . \end{equation*}
where
λ
>
0
\lambda >0
is a parameter and
f
(
u
)
=
e
u
f(u)=e^u
or
f
(
u
)
=
(
u
+
1
)
p
f(u)=(u+1)^p
where
p
>
1
p>1
and
a
(
x
)
a(x)
is a nonnegative function with certain monotonicity (we allow
a
(
x
)
=
1
a(x)=1
). Here
Ω
\Omega
is an annular domain which is also a double domain of revolution. Our interest will be in the question of the regularity of the extremal solution. We obtain improved compactness because of the annular nature of the domain and we obtain further compactness under some monotonicity assumptions on the domain.