Positive solutions for a coupled nonlinear Kirchhoff-type system with vanishing potentials

In this paper, we consider the strongly coupled nonlinear Kirchhoff-type system with vanshing potentials: [begin{cases} -left(a_1+b_1int_{mathbb{R}^3}| abla u|^2dxright)Delta u+lambda V(x)u=frac{alpha}{alpha+beta}|u|^{alpha-2}u|v|^{beta},&xinmathbb{R}^3, -left(a_2+b_2int_{mathbb{R}^3}| abla v|^2dxright)Delta v+lambda W(x)v=frac{beta}{alpha+beta}|u|^{alpha}|v|^{beta-2}v,&xinmathbb{R}^3, u,vin mathcal{D}^{1,2}(mathbb{R}^3), end{cases}] where $a_i>0$ are constants, $lambda,b_i>0$ are parameters for $i=1,2$, $alpha,beta>1$ satisfy $alpha+betale4$, the nonlinear term $F(x,u,v)=|u|^alpha|v|^beta$ is not 4-superlinear at infinity, $V(x)$, $W(x)$ are nonnegative continuous potentials. By establishing some new estimates and truncation technique, we obtain the existence of positive vector solutions for the above system when $b_1+b_2$ small and $lambda$ large. Moreover, the asymptotic behavior of these vector solutions is also explored as $textbf{b}=(b_1,b_2)to bf{0}$ and $lambdatoinfty$. In particular, our results extend some known ones in previous papers that only deals with the case where $alpha,beta>2$ with $alpha+beta<6$.