Fractional boundary value problem on the half-line

We consider the semilinear fractional boundary value problem
\begin{equation*} D^{\beta}\left(\frac{1}{b(x)}D^{\alpha}u\right)=a(x)u^{\sigma} \text{in } (0,\infty) \end{equation*}
with the conditions $\lim_{x\rightarrow 0} x^{2-\beta} \frac{1}{b(x)}D^{\alpha}u(x) =\lim_{x\rightarrow \infty} x^{1-\beta}\frac{1}{b(x)}D^{\alpha}u(x)=0$ and $\lim_{x\rightarrow 0} x^{2-\alpha}u(x)= \lim_{x\rightarrow \infty} x^{1-\alpha}u(x)=0$, where $\beta,\alpha \in (1,2)$, $\sigma\in(-1,1)$ and $D^{\beta}, D^{\alpha}$ stand for the standard Riemann–Liouville fractional derivatives. The functions $ a,b : (0,\infty)\rightarrow \mathbb{R}$ are nonnegative continuous functions satisfying some appropriate conditions. The existence and the uniqueness of a positive solution are established. Also, a description of the global behavior of this solution is given.