On periodic solutions of second-order differential equations with attractive-repulsive singularities
The paper studies the periodic problem \[ u''(t)=\frac{g(t)}{u^\mu(t)}-\frac{h(t)}{u^\lambda(t)}+f(t),\quad u(0)=u(\omega),\;u'(0)=u'(\omega), \tag{1} \] where \(g, h\in L(\mathbb{R}/\omega\mathbb{Z}; {\mathbb{R}}_+)\),\ \(f\in L(\mathbb{R}/\omega \mathbb{Z}; \mathbb{R})\), and \(\lambda, \mu >0\). Since the functions \(g\) and \(h\) are possibly zero on some sets of positive measure, the singularity may combine attractive and repulsive effects. The authors establish sufficient conditions for the existence of a solution to problem (1). Their main result can be applied to the original Lazer-Solimini equations both in the attractive and in the repulsive case. The proofs rely on the method of upper and lower functions. To illustrate the results, an application to the dynamics of a trapless Bose-Einstein condensate is given. Final sections are devoted to a comparative study of the equation with attractive (respectively repulsive) singularity. Along the paper some open problems are posed.
second-order ordinary differential equation
scientific article; zbMATH DE number 5651184