This paper is concerned with the following Gierer-Meinhardt type systems subject to Dirichlet boundary conditions:
\[
{
Δ
u
−
α
u
+
u
p
v
q
+
ρ
(
x
)
=
0
,
u
>
0
,
a
m
p
;
in
Ω
,
Δ
v
−
β
v
+
u
r
v
s
=
0
,
v
>
0
,
a
m
p
;
in
Ω
,
u
=
0
,
v
=
0
a
m
p
;
on
∂
Ω
,
\begin {cases} \Delta u - \alpha u + \frac {u^p}{v^q} + \rho (x) = 0,\; u > 0, & \text {in $\Omega $},\\ \Delta v - \beta v + \frac {u^r}{v^s} = 0,\; v > 0, & \text {in $\Omega $}, \\ u=0,\; v=0 & \text {on $\partial \Omega $}, \end {cases}
\]
where
Ω
⊂
R
N
\Omega \subset \mathbb {R}^N
(
N
≥
1
N\geq 1
) is a smooth bounded domain,
ρ
(
x
)
≥
0
\rho (x)\geq 0
in
Ω
\Omega
and
α
,
β
≥
0
\alpha ,\beta \geq 0
. We are mainly interested in the case of different source terms, that is,
(
p
,
q
)
≠
(
r
,
s
)
(p,q)\neq (r,s)
. Under appropriate conditions on the exponents
p
,
q
,
r
p,q,r
and
s
s
we establish various results of existence, regularity and boundary behavior. In the one dimensional case a uniqueness result is also presented.