Existence and global asymptotic behavior of positive solutions for nonlinear problems on the half-line

In this paper, we aim at studying the existence, uniqueness and the exact asymptotic behavior of positive solutions to the following boundary value problem{1A(Au′)′+a(t)uσ=0,t∈(0,∞),limt→0+u(t)=0,limt→∞u(t)ρ(t)=0, where σ<1, A is a continuous function on [0,∞), positive and differentiable on (0,∞) such that 1A is integrable on [0,1] and ∫0∞1A(t)dt=∞. Here ρ(t)=∫0t1A(s)ds, for t⩾0 and a is a nonnegative continuous function that is required to satisfy some assumptions related to the Karamata classes of regularly varying functions. Our arguments are based on monotonicity methods.