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 { 1 A ( A u ′ ) ′ + a ( t ) u σ = 0 , t ∈ ( 0 , ∞ ) , lim t → 0 + u ( t ) = 0 , lim t → ∞ u ( t ) ρ ( t ) = 0 , where σ < 1 , A is a continuous function on [ 0 , ∞ ) , positive and differentiable on ( 0 , ∞ ) such that 1 A is integrable on [ 0 , 1 ] and ∫ 0 ∞ 1 A ( t ) d t = ∞ . Here ρ ( t ) = ∫ 0 t 1 A ( s ) d s , 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.