A coupled system of Sturm–Liouville differential equations

The main purpose of this paper is to investigate the existence and the asymptotic behavior of positive continuous solutions of the following nonlinear coupled system:
\begin{equation*} \begin{cases} -\dfrac{1}{A}(Au')'= a(x)u^pv^r\quad\text{on}\ \ (0,1), \\[10 pt] -\dfrac{1}{B}(Bv')'=b(x)v^q u^s\quad\text{on}\ \ (0,1), \\[10 pt] u(0)=u(1)=v(0)=v(1)=0, \end{cases} \end{equation*}
where $p,q \in (-1,1)$ and $r,s \in \mathbb{R}$ are such that $ (1- |p |)(1-| q |)-|rs | >0$. The functions $A$ and $B$ are positive and differentiable on $(0,1) $, and the positive weight functions $a$ and $b$ may be singular at the boundary and satisfy some appropriate assumptions related to the Karamata class.