Fractional p -Kirchhoff problems involving critical exponents and sign-changing weight functions

Abstract

In this paper, we study the following Kirchhoff type problems involving fractional p-Laplacian and critical exponents: $$\begin{array}{c}\left\{\begin{array}{cc}\mathit{M}\mathrm{(}{[\mathit{u}]}_{\mathit{s}\mathrm{,}\mathit{p}}^{\mathit{p}}\mathrm{)}{\mathrm{(}-\mathrm{\Delta }\mathrm{)}}_{\mathit{p}}^{\mathit{s}}\mathit{u}=\mathit{\lambda }|\mathit{u}{|}^{{\mathit{p}}_{\mathit{s}}^{\ast }-2}\mathit{u}+\mathit{b}\mathrm{(}\mathit{x}\mathrm{)}|\mathit{u}{|}^{\mathit{p}-2}\mathit{u}+\mathit{f}\mathrm{(}\mathit{x}\mathrm{,}\mathit{u}\mathrm{)}\mathrm{,}\hfill & \text{in }\mathrm{\Omega }\mathrm{,}\hfill \\ \mathit{u}=0\hfill & \text{in }{\mathbb{R}}^{\mathit{N}}\setminus \mathrm{\Omega }\mathrm{,}\hfill \end{array}\right.\end{array}$$ where Ω is a bounded domain in ${\mathbb{R}}^{\mathit{N}}$ ( $\mathit{N}⩾3$ ) with Lipshcitz boundary, $1\mathrm{<}\mathit{p}\mathrm{<}\mathit{N}\mathrm{/}\mathit{s}$ with $\mathit{s}\in \mathrm{(}0\mathrm{,}1\mathrm{)}$ , ${\mathit{p}}_{\mathit{s}}^{\ast }=\mathit{N}\mathit{p}\mathrm{/}\mathrm{(}\mathit{N}-\mathit{p}\mathit{s}\mathrm{)}$ is the fractional critical Sobolev exponent, λ is a positive parameter, $\mathit{M}:[0\mathrm{,}+\infty \mathrm{)}\to {\mathbb{R}}^{+}$ and $\mathit{f}:\bar{\mathrm{\Omega }}\times \mathbb{R}\to \mathbb{R}$ are continuous functions, $\mathit{b}:\mathrm{\Omega }\to \mathbb{R}$ is a sign-changing function. By using the fractional version of concentration compactness principle together with mountain pass theorem, we obtained the multiplicity of solutions for the above problem.