Abstract
We are concerned with the existence, uniqueness and global asymptotic behavior of positive continuous solutions to the second-order boundary value problem 1A(Au′)′+a1(t)uσ1+a2(t)uσ2=0,t∈(0,∞),$\frac{1}{A}(Au^{\prime})^{\prime}+a_{1}(t)u^{\sigma_{1}}+a_{2}(t)u^{\sigma_{2}% }=0,\quad t\in(0,\infty),$ subject to the boundary conditions limt→0+u(t)=0${\lim_{t\rightarrow 0^{+}}u(t)=0}$, limt→∞u(t)/ρ(t)=0${\lim_{t\rightarrow\infty}{u(t)}/{\rho(t)}=0}$, where σ1,σ2<1${\sigma_{1},\sigma_{2}<1}$ and A is a continuous function on [0,∞)${[0,\infty)}$ which is positive and differentiable on (0,∞)${(0,\infty)}$ such that ∫011/A(t)𝑑t<∞${\int_{0}^{1}{1}/{A(t)}\,dt<\infty}$ and ∫0∞1/A(t)𝑑t=∞${\int_{0}^{\infty}{1}/{A(t)}\,dt=\infty}$. Here, ρ(t)=∫0t1/A(s)𝑑s${\rho(t)=\int_{0}^{t}{1}/{A(s)}\,ds}$ for t>0${t>0}$ and a1,a2${a_{1},a_{2}}$ are nonnegative continuous functions on (0,∞)${(0,\infty)}$ that may be singular at t=0${t=0}$ and satisfying some appropriate assumptions related to the Karamata regular variation theory. Our approach is based on the sub-supersolution method.