Nonlocal p -Kirchhoff equations with singular and critical nonlinearity terms

Abstract

The objective of this work is to investigate a nonlocal problem involving singular and critical nonlinearities: $$\begin{array}{c}\left\{\begin{array}{c}{\mathrm{(}{[\mathit{u}]}_{\mathit{s}\mathrm{,}\mathit{p}}^{\mathit{p}}\mathrm{)}}^{\mathit{\sigma }-1}{\mathrm{(}-\mathrm{\Delta }\mathrm{)}}_{\mathit{p}}^{\mathit{s}}\mathit{u}=\frac{\mathit{\lambda }}{{\mathit{u}}^{\mathit{\gamma }}}+{\mathit{u}}^{{\mathit{p}}_{\mathit{s}}^{\ast }-1}\text{in }\mathrm{\Omega }\mathrm{,}\hfill \\ \mathit{u}\mathrm{>}0\mathrm{,}\text{in }\mathrm{\Omega }\mathrm{,}\hfill \\ \mathit{u}=0\mathrm{,}\text{in }{\mathbb{R}}^{\mathit{N}}\setminus \mathrm{\Omega }\mathrm{,}\hfill \end{array}\right.\end{array}$$ where Ω is a bounded domain in ${\mathbb{R}}^{\mathit{N}}$ with the smooth boundary $\partial \mathrm{\Omega }$ , $0\mathrm{<}\mathit{s}\mathrm{<}1\mathrm{<}\mathit{p}\mathrm{<}\infty$ , $\mathit{N}\mathrm{>}\mathit{s}\mathit{p}$ , $1\mathrm{<}\mathit{\sigma }\mathrm{<}{\mathit{p}}_{\mathit{s}}^{\ast }\mathrm{/}\mathit{p}$ , with ${\mathit{p}}_{\mathit{s}}^{\ast }=\frac{\mathit{N}\mathit{p}}{\mathit{N}-\mathit{p}\mathit{s}}$ , ${\mathrm{(}-\mathrm{\Delta }\mathrm{)}}_{\mathit{p}}^{\mathit{s}}$ is the nonlocal p-Laplace operator and ${[\mathit{u}]}_{\mathit{s}\mathrm{,}\mathit{p}}$ is the Gagliardo p-seminorm. We combine some variational techniques with a truncation argument in order to show the existence and the multiplicity of positive solutions to the above problem.