Existence and concentration results for Kirchhoff-type Schr\"{o}dinger systems with steep potential well
Bull. Korean Math. Soc. 2015 Vol. 52, No. 2, 661-677
Published online March 31, 2015
Dengfeng L\"{u}
Central China Normal University
Abstract : In this paper, we consider the following Kirchhoff-type \linebreak Schr\"{o}dinger system \begin{equation} \left\{ \renewcommand{\arraystretch}{1.25} \begin{array}{ll} -\Big(a_{1}+ b_{1}\D\int_{\mathbb{R}^{3}}|\nabla u|^{2}dx\Big)\Delta u+\gamma V(x)u=\D\frac{2\alpha}{\alpha+\beta}|u|^{\alpha-2}u|v|^{\beta} & \mbox{in} \ \mathbb{R}^{3},\\[3mm] -\Big(a_{2}+ b_{2}\D\int_{\mathbb{R}^{3}}|\nabla v|^{2}dx\Big)\Delta v+\gamma W(x)v=\D\frac{2\beta}{\alpha+\beta}|u|^{\alpha}|v|^{\beta-2}v & \mbox{in} \ \mathbb{R}^{3},\\[2mm] u, \ v\in H^{1}(\mathbb{R}^{3}), \end{array}\nonumber \right.\nonumber \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.
Keywords : Kirchhoff-type Schr\"{o}dinger system, variational method, concentration, steep potential well
MSC numbers : 35J50, 35J10, 35Q60
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