  Consider the following Kirchhoff type problem $$ \left\{\aligned
-\bigg(a+b\int_{\mathbb{B}_R}|\nabla u|^2dx\bigg)\Delta u&= \lambda u^{q-1} +
\mu u^{p-1}, &\quad \text{in}\mathbb{B}_R, \\
u&>0,&\quad\text{in}\mathbb{B}_R,\\ u&=0,&\quad\text{on}\partial\mathbb{B}_R,
\endaligned \right.\eqno{(\mathcal{P})} $$ where $\mathbb{B}_R\subset
\bbr^N(N\geq3)$ is a ball, $2\leq q<p\leq2^*:=\frac{2N}{N-2}$ and $a$, $b$,
$\lambda$, $\mu$ are positive parameters. By introducing some new ideas and
using the well-known results of the problem $(\mathcal{P})$ in the cases of
$a=\mu=1$ and $b=0$, we obtain some special kinds of solutions to
$(\mathcal{P})$ for all $N\geq3$ with precise expressions on the parameters
$a$, $b$, $\lambda$, $\mu$, which reveals some new phenomenons of the solutions
to the problem $(\mathcal{P})$. It is also worth to point out that it seems to
be the first time that the solutions of $(\mathcal{P})$ can be expressed
precisely on the parameters $a$, $b$, $\lambda$, $\mu$, and our results in
dimension four also give a partial answer to Neimen's open problems [J.
Differential Equations, 257 (2014), 1168--1193]. Furthermore, our results in
dimension four seems to be almost "optimal".
