On the Existence of Positive Solutions of a Nonlinear Differential Equation

We study some existence results for the nonlinear equation (1/A)(Au')'=uψ(x,u) for x∈(0,ω) with different boundary conditions, where ω∈(0,∞], A is a continuous function on [0,ω), positive and differentiable on (0,ω), and ψ is a nonnegative function on (0,ω)×[0,∞) such that t↦tψ(x,t) is continuous on [0,∞) for each x∈(0,ω). We give asymptotic behavior for positive solutions using a potential theory approach.