Nonlocal Kirchhoff-type problems with singular nonlinearity: existence, uniqueness and bifurcation

This paper focuses on the following Kirchhoff-type problems involving fractional p-Laplacian operators and singular nonlinearities M(x,[u]s,pp)(-Δ)psu=λf(x)u-qinΩ,u>0inΩ,u=0inRN\Ω,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\left\{ \begin{array}{ll} M(x,[u]_{s,p}^p)(-\varDelta )_{p}^{s}u=\lambda f(x)u^{-q}~~\textrm{in}~~\varOmega , \\ u>0~~\textrm{in}~\varOmega ,\\ u=0~~\textrm{in}~{\mathbb {R}^N\setminus \varOmega }, \end{array}\right. } \end{aligned}$$\end{document}where s∈(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s\in (0,1)$$\end{document}, p∈(1,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p\in (1, \infty )$$\end{document}, q>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q>0$$\end{document}, ps∗=Np/(N-ps)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p^*_s=Np/(N-ps)$$\end{document} with N>ps\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N>ps$$\end{document}, M(x,t)=a(x)+b(x)t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M(x,t)=a(x)+b(x)t$$\end{document}, λ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda \in \mathbb {R}$$\end{document}. Firstly, to overcome the difficulties posed by the singularity structure and Kirchhoff term, we determine the lower bound on the weak solutions of approximating equations by super-solution techniques. Then, using this result combined with the fixed-point result, the compactness of operator is constructed, which plays an important role for bifurcation results. Meanwhile, the existence, uniqueness and L∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^\infty $$\end{document}-bound of weak solutions to the equations with singular nonlinearities are analyzed, and the strategy adopted to equation adding subcritical term. Finally, we obtain the existence of unbounded connected components bifurcating from infinite and trivial solutions by means of bifurcation theory.