Multiple solutions of fractional Kirchhoff equations involving a critical nonlinearity

Hua Jin; Wenbin Liu; Jianjun Zhang

doi:10.3934/dcdss.2018029

Article Contents

2018, Volume 11, Issue 3: 533-545. Doi: 10.3934/dcdss.2018029

This issue Previous Article Existence and multiplicity results for resonant fractional boundary value problems Next Article Optimal elliptic regularity: A comparison between local and nonlocal equations

Multiple solutions of fractional Kirchhoff equations involving a critical nonlinearity

1.
College of Science, China University of Mining and Technology, Xuzhou 221116, China
2.
College of Mathematica and Statistics, Chongqing Jiaotong University, Chongqing 400074, China

* Corresponding author: W. Liu.
* Corresponding author: W. Liu.

Received: May 2017

Revised: August 2017

Published: June 2018

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Abstract

In this paper, we are concerned with the following fractional Kirchhoff equation

$\begin{align*}\left\{\begin{array}{ll}\left(a+b∈t_{\mathbb R^N}|(-Δ)^{\frac{s}{2}}u|^2\right)(-Δ)^su=\lambda u+μ|u|^{q-2}u+|u|^{2_{s}^*-2}u&\ \ \mbox{in}\ \ Ω, \\ u=0&\ \ \mbox{in}\ \mathbb R^N\backslashΩ, \end{array}\right.\end{align*}$

where $N>2s$ , $a, b, \lambda, μ>0$ , $s∈(0, 1)$ and $Ω$ is a bounded open domain with continuous boundary. Here $(-Δ)^s$ is the fractional Laplacian operator. For $2<q≤q\min\{4, 2_s^*\}$ , we prove that if $b$ is small or $μ$ is large, the problem above admits multiple solutions by virtue of a linking theorem due to G. Cerami, D. Fortunato and M. Struwe [7, Theorem 2.5].

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Mathematics Subject Classification: Primary: 35A15, 35B33, 35J60.

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