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{equation*}\left\{\begin{array}{ll} ([u]_{s,p}^p)^{\sigma-1}(-\Delta)^s_p u = \frac{\lambda}{u^{\gamma}}+u^{ p_s^{*}-1 }\quad \text{in }\Omega,\\ u>0,\;\;\;\;\quad \text{in }\Omega,\\ u=0,\;\;\;\;\quad \text{in }\mathbb{R}^{N}\setminus \Omega,\end{array} \right. \end{equation*} where $\Omega$ is a bounded domain in $\mathbb{R}^N$ with the smooth boundary $\partial \Omega$, $0 < s< 1<p<\infty$, $N> sp$, $1<\sigma<p^*_s/p,$ with $p_s^{*}=\frac{Np}{N-ps},$ $ (- \Delta )_p^s$ is the nonlocal $p$-Laplace operator and $[u]_{s,p}$ 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.