Asymptotic behavior of positive solutions of a semilinear Dirichlet problem

Abstract

In this paper, we are concerned with the existence, uniqueness and the asymptotic behavior of a positive solution to the following semilinear differential problem \begin{equation*} -\frac{1}{A}(Au')'=q(t)u^{\sigma},\text{ \ }t\in (0,1),\text{ \ }u>0\text{ on }% (0,1) ,\text{ }u(0)=u(1)=0.% \end{equation*} Here $\sigma<1$, $q$ is a positive function in $\mathcal{C}_{loc}^{\gamma}((0,1))$, $0<\gamma<1$, satisfying an appropriate assumption related to Karamata regular variation theory and $A$ is a positive differentiable function on $(0,1)$ such that $$\displaystyle\frac{1}{c}t^{\alpha_{1}}(1-t)^{\alpha_{2}}\leq A(t)\leq ct^{\alpha_{1}}(1-t)^{\alpha_{2}},$$ where $c$ is a positive constant and $\alpha_{1},\alpha_{2}<1.$