Existence and Exact Asymptotic Behavior of Positive Solutions for a Fractional Boundary Value Problem

Abstract

We establish the existence and uniqueness of a positive solution $u$ for the fractional boundary value problem $D^{\alpha }u(x)=-a(x)u^{\sigma }(x)$, $x\in (\mathrm{0,1})$ with the condition ${\lim }_{x\to 0}{D}^{\alpha -1}u(x)=0,u(1)=0$, where , $\sigma \in (-\mathrm{1,1}),$ and $a$ is a nonnegative continuous function on $(\mathrm{0,1})$ that may be singular at $x=0$ or $x=1$.