Existence and multiple solutions for the critical fractional
$p$-Kirchhoff type problems involving sign-changing weight functions

- Jie Yang,
- Senli Liu,
- Haibo Chen

Jie Yang

Central South University School of Mathematics and Statistics

Corresponding Author:dafeyang@163.com

Author ProfileSenli Liu

Central South University School of Mathematics and Statistics

Author ProfileHaibo Chen

Department of Mathematics, Central South University

Author Profile## Abstract

The aim of this paper is to study the existence and multiplicity of nonnegative solutions for the following critical Kirchhoff equation
involving the fractional \(p\)-Laplace operator \((-\Delta)_{p}^{s}\). More precisely, we consider

\begin{equation}
\begin{cases}M\left(\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}dxdy\right)(-\Delta)_{p}^{s}u=\lambda f(x)|u|^{q-2}u+K(x)|u|^{p_{s}^{*}-2}u,&{\rm in}\ \Omega,\\
u=0,&{\rm in}\ \mathbb{R}^{N}\setminus\Omega,\\
\end{cases}\nonumber \\
\end{equation}
where \(\Omega\subset\mathbb{R}^{N}\)
is an open bounded domain with Lipschitz boundary \(\partial\Omega\), \(M(t)=a+bt^{m-1}\) with \(m>1,a>0,b>0\),
dimension \(N>sp\), \(p_{s}^{*}=\frac{Np}{N-ps}\) is the fractional critical Sobolev exponent, and the parameters
\(\lambda>0,0<s<1<q<p<\infty\). Applying Nehari manifold, fibering maps and Krasnoselskii genus theory, we investigate the existence and multiplicity
of nonnegative solutions.

01 Mar 2020Submitted to *Mathematical Methods in the Applied Sciences* 07 Mar 2020Submission Checks Completed

07 Mar 2020Assigned to Editor

10 Jul 2020Reviewer(s) Assigned

19 Oct 2021Editorial Decision: Revise Minor

28 Oct 20211st Revision Received

28 Oct 2021Submission Checks Completed

28 Oct 2021Assigned to Editor

28 Oct 2021Reviewer(s) Assigned

08 Feb 2022Review(s) Completed, Editorial Evaluation Pending

09 Feb 2022Editorial Decision: Accept