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We consider a relativistic particle under the action of a time-periodic central force field in the plane. When it is attractive at a given level there are many subharmonic and quasiperiodic motions.

We study the existence of positive solutions for the model scalar second-order boundary value problem ⎧ ⎩ −u �� + c(x)u � + a(x)u = b(x) u(x)p ,x ∈ R, lim|x|→∞ u(x )=0 , where a,b,c are locally bounded coefficients and p> 0.

In this paper, we study the existence of positive periodic solutions to second-order singular differential equations. The proof relies on Schauder’s fixed point theorem. Our results show that in some situations weak singularities can help create periodic solutions, as pointed out by Torres [J. Differential Equations 232 (2007) 277–284]. In this paper, we study the existence of positive periodic solutions of the second-order differential equation x �� + a(t)x = f (t, x )+ e(t); (1.1) here, a(t) and e(t) are continuous and 1-periodic functions. The nonlinearity f (t, x )i s continuous in (t, x) and 1-periodic in t. We are mainly interested in the case that f (t, x) may be singular at x =0 . Beginning with the paper of Lazer and Solimini [10], the semilinear singular differential equation