Existence and multiplicity of positive periodic solutions to a class of Liénard equations with repulsive singularities

Abstract

The purpose of this paper is to study the non-existence, existence and multiplicity of positive T-periodic solutions to the following parameter-dependent Liénard equations with repulsive singularities

$$\begin{aligned} x''+f(x)x'-\frac{a(t)}{x^\mu }+\varphi (t)x^\delta =s, \end{aligned}$$

where $f:(0,+\infty )\rightarrow \mathbb {R},$ $a\in C[0,T]$ is a nonnegative function with $\overline{a}>0,$ $\varphi \in C[0,T]$ is a sign-changing function with $\overline{\varphi }<0,$ constants $\mu >0,~\delta \in (0,1)$ are fixed, and $s\in \mathbb {R}$ is a parameter. Using a continuation theorem of coincidence degree theory and certain properties of Leray–Schauder degree, we prove the existence of critical point $s_*\in \mathbb {R}$ such that the equation has at least two, at least one or no positive T-periodic solution, according to $s<s_*,$ $s=s_*$ or $s>s_*,$ respectively.