In this paper we study the existence of positive solutions for problems of the type
−
Δ
p
u
(
x
)
a
m
p
;
=
u
(
x
)
q
−
1
h
(
x
,
u
(
x
)
)
,
a
m
p
;
a
m
p
;
x
∈
Ω
,
u
(
x
)
a
m
p
;
=
0
,
a
m
p
;
a
m
p
;
x
∈
∂
Ω
,
\begin{equation*} \begin {aligned} -\Delta _pu(x) &=u(x)^{q-1}h(x,u(x)), && x\in \Omega , \\ u(x)&=0, && x\in \partial \Omega , \end{aligned} \end{equation*}
where
Δ
p
\Delta _p
is the
p
p
-Laplace operator and
p
,
q
>
1
p,q>1
. If
p
=
2
p=2
, such problems arise in population dynamics. Making use of different methods (sub- and super-solutions and a variational approach), we treat the cases
p
=
q
p=q
,
p
>
q
p>q
and
p
>
q
p>q
, respectively. Also, some systems of equations are considered.