2\(\pi\)-periodic self-similar solutions for the anisotropic affine curve shortening problem (Q543394)

The study of the generalized curve shortening problem motivates the content of this paper, where the authors consider the existence of \(n\pi\)-periodic (\(n \geq 2\)) positive solutions of the equation \[ u_{\theta \theta} + u = \frac{a(\theta)}{u^3}. \] Here, \(a(\theta)\) is a positive and smooth \(n\pi\)-periodic function. There is a special emphasis on the case \(n=2.\) In the proof, the authors carry out a careful analysis of the interaction between different blow-ups, the Lyapunov-Schmidt reduction and degree theory are also used.