Periodic solutions for some second order differential equations with singularity (Q943861)

The paper deals with the second-order singular differential equation \[ x''(t)+g(x)=a(t)x^\gamma+b(t,x), \leqno(1) \] where \(g\) is continuous on \((0,\infty)\), superlinear near infinity, and has a strong singularity at \(x=0\), \(a\) is continuous and \(2\pi\)-periodic, \(b\) is continuous, \(2\pi\)-periodic in its first variable and bounded, \(\gamma\in [0,1]\). The authors prove that under these conditions equation (1) has infinitely many positive \(2\pi\)-periodic solutions \(x_j\) and infinitely many positive subharmonic solutions \(y_j\) and that their \(C^1\)-sup norms go to infinity for \(j\to \infty\). The proofs are based on the phase plane analysis methods and the Poincaré-Birkhoff theorem.