International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 58658, 12 pages
doi:10.1155/2007/58658

On the existence of positive solutions of a nonlinear differential equation

Abstract

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.