| Abstract: | This paper is concerned with the following Kirchhoff-type equations$$left{begin{array}{ll}displaystyle-big(varepsilon^{2}a+varepsilon bint_{mathbb{R}^{3}}|nabla u|^{2}mathrm{d}xbig)Delta u+ V(x)u+muphi |u|^{p-2}u=f(x,u), &quad mbox{ in }mathbb{R}^{3},(-Delta)^{frac{alpha}{2}} phi=mu|u|^{p},~u>0, &quad mbox{ in }mathbb{R}^{3},end{array}right.$$where $f(x,u)=lambda K(x)|u|^{q-2}u+Q(x)|u|^{4}u$, $a>0,~b,~mugeq0$ are constants, $alphain(0,3)$, $pin[2,3),~qin[2p,6)$ and $varepsilon,~lambda>0$ are parameters. Under some mild conditions on $V(x),~K(x)$ and $Q(x)$, we prove that the above system possesses a ground state solution $u_{varepsilon}$ with exponential decay at infinity for $lambda>0$ and $varepsilon$ small enough. Furthermore, $u_{varepsilon}$ concentrates around a global minimum point of $V(x)$ as $varepsilonrightarrow0$. The methods used here are based on minimax theorems and the concentration-compactness principle of Lions. Our results generalize and improve those in Liu and Guo (Z Angew Math Phys 66: 747-769, 2015), Zhao and Zhao (Nonlinear Anal 70: 2150-2164, 2009) and some other related literature. |
