A note on the existence of solutions with prescribed asymptotic behavior for half-linear ordinary differential equations

Abstract

The half-linear differential equation (|u'|^{\alpha}{\rm sgn} u')' = \alpha(\lambda^{\alpha+ 1} + b(t))|u|^{\alpha}{\rm sgn} u, \quad t \geq t_0, is considered, where $\alpha$ and $\lambda$ are positive constants and $b(t)$ is a real-valued continuous function on $[t_0,\infty)$. It is proved that, under a mild integral smallness condition of $b(t)$ which is weaker than the absolutely integrable condition of $b(t)$, the above equation has a nonoscillatory solution $u_0(t)$ such that $u_0(t) \sim{\rm e}^{- \lambda t}$ and $u_0'(t) \sim- \lambda{\rm e}^{- \lambda t}$ ($t \to\infty$), and a nonoscillatory solution $u_1(t)$ such that $u_1(t) \sim{\rm e}^{\lambda t}$ and $u_1'(t) \sim\lambda{\rm e}^{\lambda t}$ ($t \to\infty$).

Keywords

• half-linear differential equation
• nonoscillatory solution
• asymptotic form