On the Existence of Positive Solutions of a Nonlinear Differential Equation

We study some existence results for the nonlinear equation ( 1 / A ) ( A u ' ) ' = u ψ ( 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 ↦ t ψ ( x , t ) is continuous on [ 0 , ∞ ) for each x ∈ ( 0 , ω ) . We give asymptotic behavior for positive solutions using a potential theory approach.