  We are concerned with a class of Kirchhoff type equations in $\mathbb{R}^{N}$
as follows: \begin{equation*} \left\{ \begin{array}{ll} -M\left(
\int_{\mathbb{R}^{N}}|\nabla u|^{2}dx\right) \Delta u+\lambda V\left( x\right)
u=f(x,u) & \text{in }\mathbb{R}^{N}, \\ u\in H^{1}(\mathbb{R}^{N}), &
\end{array}% \right. \end{equation*}% where $N\geq 1,$ $\lambda>0$ is a
parameter, $M(t)=am(t)+b$ with $a,b>0$ and $m\in
C(\mathbb{R}^{+},\mathbb{R}^{+})$, $V\in C(\mathbb{R}^{N},\mathbb{R}^{+})$ and
$f\in C(\mathbb{R}^{N}\times \mathbb{R}, \mathbb{R})$ satisfying
$\lim_{|u|\rightarrow \infty }f(x,u) /|u|^{k-1}=q(x)$ uniformly in $x\in
\mathbb{R}^{N}$ for any $2<k<2^{\ast}$($2^{\ast}=\infty$ for $N=1,2$ and
$2^{\ast}=2N/(N-2)$ for $N\geq 3$). Unlike most other papers on this problem,
we are more interested in the effects of the functions $m$ and $q$ on the
number and behavior of solutions. By using minimax method as well as
Caffarelli-Kohn-Nirenberg inequality, we obtain the existence and multiplicity
of positive solutions for the above problem.
