Positive solutions for asymptotically periodic Kirchhoff-type equations with critical growth

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In this paper‎, ‎we consider the following Kirchhoff-type equations‎: ‎$-‎left(a+bint_{mathbb{R}^{3}}|nabla u|^{2}right)Delta u+V(x) u=lambda$ $f(x,u)+u^{5}‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$u(x)>0‎, ‎quad mbox{in }mathbb{R}^{3},$ ‎$uin H^{1}(mathbb{R}^{3})‎ ,‎$ ‎ ‎‎‎where $a,b>0$ are constants and $lambda$ is a positive parameter‎. ‎The aim of this paper is to study the existence of positive solutions for Kirchhoff-type equations with a nonlinearity in the critical growth under some suitable assumptions on $V(x)$ and $f(x,u)$‎. ‎Recent results from the literature are improved and extended.