In this paper, the authors analyze the existence, multiplicity and dynamics of \(T\)-periodic solutions of the Liénard equation \[ x''+f(x)x'+h(t,x)=s, \] where \(f:(0,+\infty)\rightarrow\mathbb{R}\) is continuous, \(h:\mathbb{R}\times(0,+\infty)\rightarrow\mathbb{R}\) is continuous, \(T\)-periodic in the first variable and has a repulsive singularity at the origin, and \(s\) is a real parameter. Moreover, some boundedness and asymptotic assumptions on \(h\) and on the primitive function of \(f\) are made.
The main result of the paper is a classical Ambrosetti-Prodi type result for the equation above. Namely, it is proven the existence of a \(s^\ast\in\mathbb{R}\) such that the equation has \(T\)-periodic positive solutions if, and only if, \(s\leq s^\ast\). More precisely, it has, at least, one if \(s=s^\ast\) and, at least, two if \(s<s^\ast\). Furthermore, when \(s\to-\infty\), there is a \(T\)-periodic positive solution whose minimum tends to \(+\infty\) and another one whose minimum tends to 0.
Finally, some particulars cases and an application to indefinite problems are studied.