A General Result to the Existence of a Periodic Solution to an Indefinite Equation with a Weak Singularity

Abstract

Efficient conditions guaranteeing the existence of a T-periodic solution to the second order differential equation $$\begin{aligned} u''=\frac{h(t)}{u^{\lambda }},\quad \lambda \in (0,1), \end{aligned}$$ are established. Here, $$h\in L(\mathbb {R}/T\mathbb {Z})$$ is a rather general sign-changing function with $$\overline{h}<0$$ . In contrast with the results in Godoy and Zamora (Proc R Soc Edinb Sect A Math) and Hakl and Zamora (J Differ Equ 263:451–469, 2017), the key ingredient to solve the aforementioned problem seems to be connected more with the oscillation and the symmetry aspects of the weight function h than with the multiplicity of its zeroes. Roughly speaking, the solvability for the above-mentioned problem can be guaranteed when $$H_+\approx H_-$$ and $$H_+$$ is large enough.