Existence and exact asymptotic behaviour of positive solutions for fractional boundary value problem with P-Laplacian operator

This paper deals with existence, uniqueness and global behaviour of a positive solution for the fractional boundary value problem $D^{\beta }(\psi (x)\Phi _{p}(D^{\alpha }u))=a(x)u^{\sigma }$ in $(0,1)$ with the condition $$\begin{equation*} \underset{x\rightarrow 0}\lim D^{\beta-1}(\psi(x)\Phi_{p}(D^{\alpha}u(x) ))=\underset{x\rightarrow 1}\lim \psi(x)\Phi_{p}(D^{\alpha}u(x))=0 \quad {\rm and} \quad \underset{x\rightarrow 0}\lim D^{\alpha-1}u(x)= u(1)=0, \end{equation*}$$ where $\beta ,\alpha \in (1,2]$ , $\Phi _{p}(t)=t|t|^{p-2}$ , p>1, $\sigma \in (1-p,p-1)$ , the differential operator is taken in the Riemann–Liouville sense and $\psi , a\ : (0,1)\longrightarrow \mathbb {R}$ are non-negative and continuous functions that may are singular at x=0 or x=1 and satisfies some appropriate conditions