Ground states for Kirchhoff-type equations with critical growth

Quanqing Li; Kaimin Teng; Xian Wu

doi:10.3934/cpaa.2018124

Article Contents

2018, Volume 17, Issue 6: 2623-2638. Doi: 10.3934/cpaa.2018124

This issue Previous Article The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior Next Article Positive solutions for resonant (p, q)-equations with concave terms

Ground states for Kirchhoff-type equations with critical growth

1.
Department of Mathematics, Honghe University, Mengzi, Yunnan 661100, China
2.
Department of Mathematics, Tsinghua University, Beijing, Beijing 100084, China
3.
Department of Mathematics, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China
4.
Department of Mathematics, Yunnan Normal University, Kunming, Yunnan 650092, China

Received: December 31, 2016

Revised: June 30, 2017

Published: June 2018

This work is supported in part by the National Natural Science Foundation of China (11501403; 11461023; 11701322; 11561072) and the Shanxi Province Science Foundation for Youths under grant 2013021001-3 and the Honghe University Doctoral Research Programs (XJ17B11, XJ17B12) and the Yunnan Province Local University (Part) Basic Research Joint Project (2017FH001-013).

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Abstract

In this paper, we study the following Kirchhoff-type equation with critical growth

$-(a+b\int {_{\mathbb{R}^3}} |\nabla u|^2dx)\triangle u+V(x)u = λ f(x,u)+|u|^4u, \; x \; ∈\mathbb{R}^3,$

where a>0, b>0, λ>0 and f is a continuous superlinear but subcritical nonlinearity. When V and f are asymptotically periodic in x, we prove that the equation has a ground state solution for large λ by Nehari method. Moreover, we regard b as a parameter and obtain a convergence property of the ground state solution as $b\searrow 0$.

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Mathematics Subject Classification: 35J20, 35J70, 35P05, 35P30, 34B15, 58E05, 47H04.

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