Ground State Solutions of Schrödinger-Kirchhoff Equations with Potentials Vanishing at Infinity

Abstract

In this paper, we deal with the following Schrödinger-Kirchhoff equation with potentials vanishing at infinity: $-\left({\epsilon }^{2}a+\epsilon b{\int }_{{ℝ}^3}{\left|\nabla u\right|}^2\right)\Delta u+V\left(x\right)u=K\left(x\right){\left|u\right|}^{p-1}u\text{in}{ℝ}^3$ and $u>0,u\in H^1\left({ℝ}^3\right),$ where $V\left(x\right)~{\left|x\right|}^{-\alpha }$ and $K\left(x\right)~{\left|x\right|}^{-\beta }$ with $0<\alpha <2$ , and $\beta >0$ . We first prove the existence of positive ground state solutions $u_{\epsilon }\in H^1\left({ℝ}^3\right)$ under the assumption that $\sigma <p<5$ for some $\sigma ={\sigma }_{\alpha ,\beta }$ , then we show that $u_{\epsilon }$ concentrates at a global minimum point of $\mathcal{A}\left(x\right)=V^{2/\left(p-1\right)-1/2}\left(x\right)/K^{2/\left(p-1\right)}\left(x\right)$ .