The effect of nonlocal term on the superlinear Kirchhoff type equations in $mathbb{R}^{N}$

We are concerned with a class of Kirchhoff type equations in $mathbb{R}^{N}$ as follows: begin{equation*} left{ begin{array}{ll} -Mleft( int_{mathbb{R}^{N}}| abla u|^{2}dxright) Delta u+lambda Vleft( xright) u=f(x,u) & text{in }mathbb{R}^{N}, uin H^{1}(mathbb{R}^{N}), & end{array}% right. end{equation*}% where $Ngeq 1,$ $lambda>0$ is a parameter, $M(t)=am(t)+b$ with $a,b>0$ and $min C(mathbb{R}^{+},mathbb{R}^{+})$, $Vin C(mathbb{R}^{N},mathbb{R}^{+})$ and $fin C(mathbb{R}^{N}times mathbb{R}, mathbb{R})$ satisfying $lim_{|u|rightarrow infty }f(x,u) /|u|^{k-1}=q(x)$ uniformly in $xin mathbb{R}^{N}$ for any $2<k<2^{ast}$($2^{ast}=infty$ for $N=1,2$ and $2^{ast}=2N/(N-2)$ for $Ngeq 3$). 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.