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

Abstract

In this paper the half-linear differential equation with one-dimensional p-Laplacian (p=α+1),(|u′|αsgnu′)′=α(λα+1+b(t))|u|αsgnu,t≥t0, is considered, where α and λ are positive constants. It is proved that if the function b(t) is absolutely integrable on [t0,∞), then the above equation has two nonoscillatory solutions u+(t) and u−(t) such that u±(t)∼ce±λt and u±′(t)∼±cλe±λt (t→∞), and moreover, any nontrivial solution u(t) of the above equation satisfies either u(t)∼ceλt, u′(t)∼cλeλt (t→∞) or u(t)∼ce−λt, u′(t)∼−cλe−λt (t→∞). Here, c is a nonzero constant. A weaker result is also shown under the weaker condition that b(t) is conditionally integrable on [t0,∞).