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