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} -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 $N\ge 1,$ $\lambda >0$ 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)$uniformlyin$x\in $\mathbb{R}$^{N}$forany$2<k<2^{\ast}$($2^{\ast}=\infty$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.