We discuss the properties of the differential equation u ′′ ( t ) = ( a / t ) u ′ ( t ) + f ( t , u ( t ) , u ′ ( t ) ) , a.e. on ( 0 , T ] , where a ∈ ℝ \ { 0 } , and f satisfies the L p -Carathéodory conditions on [ 0 , T ] × ℝ 2 for some p > 1 . A full description of the asymptotic behavior for t → 0 + of functions u satisfying the equation a.e. on ( 0 , T ] is given. We also describe the structure of boundary conditions which are necessary and sufficient for u to be at least in C 1 [ 0 , T ] . As an application of the theory, new existence and/or uniqueness results for solutions of periodic boundary value problems are shown.