Least energy solutions for fractional Kirchhoff type equations involving critical growth

Yinbin Deng; Wentao Huang

doi:10.3934/dcdss.2019126

Article Contents

2019, Volume 12, Issue 7: 1929-1954. Doi: 10.3934/dcdss.2019126

This issue Previous Article Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems Next Article Branching and bifurcation

Least energy solutions for fractional Kirchhoff type equations involving critical growth

Yinbin Deng^1,2, , and
Wentao Huang^3,

1.
School of Mathematics and information, Guangxi University, Nanning 530004, China
2.
Department of Mathematics, Central China Normal University, Wuhan 430079, China
3.
School of Science, East China JiaoTong University, Nanchang 330013, China

^* Corresponding author: Yinbin Deng
^* Corresponding author: Yinbin Deng

Received: October 31, 2017

Revised: March 31, 2018

Published: June 2019

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Abstract

We study the following fractional Kirchhoff type equation:

$ \begin{equation*} \begin{array}{ll} \left \{ \begin{array}{ll} \Big(a+b\int_{ \mathbb{R} ^3}|(-\Delta)^\frac{s}{2}u|^2dx\Big)(-\Delta )^s u+V(x)u = f(u)+|u|^{2^*_s-2}u, \ x\in \mathbb{R} ^3, \\ u\in H^s( \mathbb{R} ^3), \end{array} \right . \end{array} \end{equation*} $

where $ a, \ b>0 $ are constants, $ 2^*_s = \frac{6}{3-2s} $ with $ s\in(0, 1) $ is the critical Sobolev exponent in $ \mathbb{R} ^3 $, $ V $ is a potential function on $ \mathbb{R} ^3 $. Under some more general assumptions on $ f $ and $ V $, we prove that the given problem admits a least energy solution by using a constrained minimization on Nehari-Pohozaev manifold and monotone method.

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Mathematics Subject Classification: Primary: 35R11; Secondary: 35A15, 35B33.

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