Multiple Positive Periodic Solutions to Minkowski-Curvature Equations with a Singularity of Attractive Type

Abstract

Using Leray–Schauder degree theory, we investigate the existence and multiplicity of positive T-periodic solutions to Minkowski-curvature equations with a singularity of attractive type

$$\begin{aligned} \left( \frac{u'}{\sqrt{1-u'^{2}}}\right) '+f(u)u'+\varphi (t)u^{m}+\frac{r(t)}{u^\mu }=s, \end{aligned}$$

where $f\in C((0,+\infty ),\mathbb {R})$, $\varphi \in C(\mathbb {R},\mathbb {R})$ and $r\in C(\mathbb {R},(0,+\infty ))$ are T-periodic functions, m and $\mu$ are two positive constants and $0<m\le 1$, $s\in \mathbb {{R}}$ is a parameter. Based on a new method to construct upper and lower functions and some properties of Leray–Schauder degree, a multiplicity result of Ambrosetti-Prodi type is established.