Conditions are formulated which guarantee the existence of positive solutions for systems of the form
\[
−
Δ
u
1
+
f
1
(
u
1
,
…
,
u
n
)
=
μ
1
,
−
Δ
u
2
+
f
2
(
u
1
,
…
,
u
n
)
=
μ
2
,
⋮
⋮
⋮
⋮
−
Δ
u
n
+
f
n
(
u
1
,
…
,
u
n
)
=
μ
n
,
\begin {gathered} - \Delta {u_1} + {f_1}({u_1}, \ldots ,\,{u_n}) = {\mu _1}, \hfill \\ - \Delta {u_2} + {f_2}({u_1}, \ldots ,\,{u_n}) = {\mu _2}, \hfill \\ \vdots \quad \quad \quad \quad \quad \vdots \quad \quad \quad \vdots \quad \vdots \hfill \\ - \Delta {u_n} + {f_n}({u_1}, \ldots ,\,{u_n}) = {\mu _n}, \hfill \\ \end {gathered}
\]
, where
Δ
\Delta
is the Laplacian (with Dirichlet boundary conditions) on an open domain in
R
d
{\mathbf {R}^d}
, and where each
μ
i
{\mu _i}
is a positive measure. The main tools used are probabilistic potential theory, Markov processes, and an iterative scheme which is not a generalization of the one used for quasimonotone systems. Quasimonotonicity is not assumed and new results are obtained even for the case where
∂
f
k
/
∂
x
j
>
0
\partial {f_k}/\partial {x_j} > 0
for every
k
k
and
j
j
.