Nonlocal $p$-Kirchhoff equations with singular and critical nonlinearity terms

TL;DR

This paper studies a nonlocal p-Kirchhoff problem with singular and critical nonlinearities, establishing existence and multiplicity of positive solutions using variational methods and truncation techniques.

Contribution

It introduces new variational approaches to handle the combined singular and critical nonlinearities in nonlocal p-Kirchhoff equations.

Findings

Existence of positive solutions proven.

Multiple solutions established under certain conditions.

Application of variational and truncation methods to complex nonlinear problems.

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.