Existence and boundary behavior of positive solution for a Sturm–Liouville fractional problem with p-laplacian

We take up the existence and uniqueness of a positive solution for the following Sturm–Liouville boundary value problem of fractional differential equation with p-Laplacian Dβ(ρ(x)Φp(Dαu))=a(x)uσ,x∈(0,1),limx→0x2-βρ(x)Φp(Dαu(x))=limx→1Dαu(x)=0,limx→0x2-αu(x)=u(1)=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} D^{\beta }(\rho (x)\Phi _{p}(D^{\alpha }u))=a(x)u^{\sigma }, \quad x\in (0,1), \\ \underset{x\rightarrow 0}{\lim }x^{2-\beta }\rho (x)\Phi _{p}(D^{\alpha }u(x) )\,{=}\, \underset{x\rightarrow 1}{\lim }D^{\alpha }u(x)\,{=}\,0, \quad \underset{x\rightarrow 0}{\lim }x^{2-\alpha }u(x)= u(1)\,{=}\,0, \end{array} \right. \end{aligned}$$\end{document}where β,α∈(1,2]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta ,\alpha \in (1,2]$$\end{document}, Φp(t)=t|t|p-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi _{p}(t)=t|t|^{p-2}$$\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}$$p>1$$\end{document}, σ∈(1-p,p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in (1-p,p-1)$$\end{document}, Dα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{\alpha }$$\end{document} and Dβ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$D^{\beta }$$\end{document} stand for the standard Riemann–Liouville fractional derivatives. Here ρ,a:(0,1)⟶R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho , a\ : (0,1)\longrightarrow \mathbb {R}$$\end{document} are positive and continuous functions that may be singular at x=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x = 0$$\end{document} or x=1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x = 1$$\end{document} and satisfy some appropriate conditions. We also give the global behavior of a such solution.