Consider the Kirchhoff type problem
{
−
(
a
+
b
∫
B
R
|
∇
u
|
2
d
x
)
Δ
u
a
m
p
;
=
λ
u
q
−
1
+
μ
u
p
−
1
,
a
m
p
;
a
m
p
;
in
B
R
,
u
a
m
p
;
>
0
,
a
m
p
;
a
m
p
;
in
B
R
,
u
a
m
p
;
=
0
,
a
m
p
;
a
m
p
;
on
∂
B
R
,
\begin{equation}\tag {$\mathcal {P}$} \left \{ \begin {aligned} -\bigg (a+b\int _{\mathbb {B}_R}|\nabla u|^2dx\bigg )\Delta u&= \lambda u^{q-1} + \mu u^{p-1}, &&\text {in $\mathbb {B}_R$}, \\ u&>0, &&\text {in $\mathbb {B}_R$},\\ u&=0, &&\text {on $\partial \mathbb {B}_R$}, \end{aligned} \right . \end{equation}
where
B
R
⊂
R
N
(
N
≥
3
)
\mathbb {B}_R\subset \mathbb {R}^N(N\geq 3)
is a ball,
2
≤
q
>
p
≤
2
∗
:=
2
N
N
−
2
2\leq q>p\leq 2^*:=\frac {2N}{N-2}
and
a
a
,
b
b
,
λ
\lambda
,
μ
\mu
are positive parameters. By introducing some new ideas and using the well-known results of the problem
(
P
)
(\mathcal {P})
in the cases of
a
=
μ
=
1
a=\mu =1
and
b
=
0
b=0
, we obtain some special kinds of solutions to
(
P
)
(\mathcal {P})
for all
N
≥
3
N\geq 3
with precise expressions on the parameters
a
a
,
b
b
,
λ
\lambda
,
μ
\mu
, which reveals some new phenomenons of the solutions to the problem
(
P
)
(\mathcal {P})
. It is also worth pointing out that it seems to be the first time that the solutions of
(
P
)
(\mathcal {P})
can be expressed precisely on the parameters
a
a
,
b
b
,
λ
\lambda
,
μ
\mu
, and our results in dimension four also give a partial answer to Naimen’s open problems [J. Differential Equations 257 (2014), 1168–1193]. Furthermore, our results in dimension four seem to be almost “optimal”.