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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
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Senli Liu
Central South University School of Mathematics and Statistics
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Haibo Chen
Department of Mathematics, Central South University
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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.

Peer review status:UNDER REVIEW

01 Mar 2020Submitted to Mathematical Methods in the Applied Sciences
07 Mar 2020Assigned to Editor
07 Mar 2020Submission Checks Completed
10 Jul 2020Reviewer(s) Assigned