On the positive periodic solutions of a class of Liénard equations with repulsive singularities in degenerate case

Abstract

In this paper, we study the existence, multiplicity and dynamics of positive periodic solutions to a generalized Liénard equation with repulsive singularities. The Ambrosetti-Prodi type result is proved in the absence of the so-called anticoercivity condition. Furthermore, with s as a parameter, under some conditions on the function h, it has been shown that for any M>1 there exists sM∈R such that the equation x″+f(x)x′+h(t,x)=s has two positive T-periodic solutions u1(⋅;s) and u2(⋅;s) satisfying min⁡{u1(t;s):t∈[0,T]}>M and min⁡{u2(t;s):t∈[0,T]}<1/M for every s<sM. As a by-product of the property, we obtain sufficient conditions to guarantee the existence of positive T-periodic solutions of indefinite differential equations.