On a singular periodic Ambrosetti-Prodi problem (Q346609)

The equation under study is NEWLINE\[NEWLINEx''+f(x)x'+h(t,x)=s,\quad a<x<b,NEWLINE\]NEWLINE where \(f\) and \(h\) are continuous and \(h\) is \(T\)-periodic in \(t\). The main assumption is the coercivity of \(h\) with respect to \(x\), NEWLINE\[NEWLINEh(t,x)\to +\infty \text{ as } x\to a^+ \text{ or } x\to b^- NEWLINE\]NEWLINE uniformly in \(t\). When \(a=-\infty\), \(b=+\infty\), \textit{C. Fabry} et al. [Bull. Lond. Math. Soc. 18, 173--180 (1986; Zbl 0586.34038)] proved the existence of a real number \(s_0\) such that the equation has no \(T\)-periodic solution if \(s<s_0\), at least one if \(s=s_0\) and at least two if \(s>s_0\). In the present paper this result is extended to equations with singularities, meaning that \(a\) or \(b\) can be finite. The case \(b<+\infty\) is the most delicate and requires some minor additional assumptions on \(f\) and \(h\). To illustrate the result the authors introduce an interesting model for the motion of a charged particle in a class of electric fields.NEWLINENEWLINEThe second part of the paper is devoted to obtain related results for the equation NEWLINE\[NEWLINE\rho ''-\frac{\mu^2}{\rho^2} +h(t,\rho )=s,NEWLINE\]NEWLINE appearing in certain perturbations of Kepler problem.