Existence and concentration results for Kirchhoff-type Schr

摘要

In this paper, we consider the following Kirchhoff-type linebreak Schr"{o}dinger system egin{equation} left{ enewcommand{rraystretch}{1.25} egin{array}{ll} -Big(a_{1}+ b_{1}Dint_{mathbb{R}^{3}}|abla u|^{2}dxBig)Delta u+gamma V(x)u=Drac{2lpha}{lpha+eta}|u|^{lpha-2}u|v|^{eta} & mbox{in} mathbb{R}^{3},[3mm] -Big(a_{2}+ b_{2}Dint_{mathbb{R}^{3}}|abla v|^{2}dxBig)Delta v+gamma W(x)v=Drac{2eta}{lpha+eta}|u|^{lpha}|v|^{eta-2}v & mbox{in} mathbb{R}^{3},[2mm] u, vin H^{1}(mathbb{R}^{3}), end{array}onumber ight.onumber end{equation} where $a_{i}$ and $b_{i}$ are positive constants for $i=1,2$, $gamma>0$ is a parameter, $V(x)$ and $W(x)$ are nonnegative continuous potential functions. By applying the Nehari manifold method and the concentration-compactness principle, we obtain the existence and concentration of ground state solutions when the parameter $gamma$ is sufficiently large.