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

Document Type : Research Paper

Authors

School of Mathematics and Statistics‎, ‎Central South University‎, ‎Changsha‎, ‎410083 Hunan‎, ‎P‎. ‎R‎. ‎China.

Abstract

In this paper‎, ‎we consider the following Kirchhoff-type equations‎:
‎$-‎\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)\Delta u+V(x) u=\lambda$ $f(x,u)+u^{5}‎, ‎\quad \mbox{in }\mathbb{R}^{3},$
‎$u(x)>0‎, ‎\quad \mbox{in }\mathbb{R}^{3},$
‎$u\in H^{1}(\mathbb{R}^{3})‎ ,‎$ ‎
‎‎‎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.

Keywords

Main Subjects

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### History

• Receive Date: 08 May 2015
• Revise Date: 10 September 2015
• Accept Date: 22 October 2015