POSITIVE SOLUTIONS FOR FRACTIONAL BOUNDARY VALUE PROBLEM WITH LAPLACIAN OPERATOR: EXISTENCE AND ASYMPTOTIC BEHAVIOR

Abstract

This paper deals with existence and uniqueness of a positive solution for the fractional boundary value problem $ D^{\beta}(\rho(x)D^{\alpha}u)=a(x)u^{\sigma}$ in $(0,1)$ with the condition \begin{equation*} \underset{x\rightarrow 0}\lim D^{\beta-1}(\rho(x)D^{\alpha}u(x) )=\underset{x\rightarrow 1}\lim \rho(x)D^{\alpha}u(x)=0\text{ and }\underset{x\rightarrow 0}\lim D^{\alpha-1}u(x)= u(1)=0, \end{equation*} where $\beta,\alpha \in (1,2]$, $\sigma\in(-1,1)$, the differential operator is taken in the Riemann-Liouville sense and $\rho, a\ : (0,1)\longrightarrow \mathbb{R}$ are nonnegative and continuous functions that may are singular at $x = 0$ or $x = 1$ and satisfies some appropriate conditions. We also give the global behavior of a such solution.