On the existence and asymptotic behavior of solutions of half-linear ordinary differential equations.

In this paper the half-linear differential equation with one-dimensional p -Laplacian (p = α + 1), (| u ′ | α sgn u ′) ′ = α (λ α + 1 + b (t)) | u | α sgn u , t ≥ t 0 , is considered, where α and λ are positive constants. It is proved that if the function b (t) is absolutely integrable on [ t 0 , ∞) , then the above equation has two nonoscillatory solutions u + (t) and u − (t) such that u ± (t) ∼ c e ± λ t and u ± ′ (t) ∼ ± c λ e ± λ t (t → ∞), and moreover, any nontrivial solution u (t) of the above equation satisfies either u (t) ∼ c e λ t , u ′ (t) ∼ c λ e λ t (t → ∞) or u (t) ∼ c e − λ t , u ′ (t) ∼ − c λ e − λ t (t → ∞). Here, c is a nonzero constant. A weaker result is also shown under the weaker condition that b (t) is conditionally integrable on [ t 0 , ∞). [ABSTRACT FROM AUTHOR]