Remarks on a planar conformal curvature problem

Abstract

Given \(p<0\) and a positive \(L^{1}\) function \(g\), with \(g(\theta +T)=g(\theta )\) for some \(T\le 2\pi \), it is shown that the functional \({\mathcal {I}}_{T}(u)=\frac{1}{2}\int _{S}u^{2}+u'^{2}d\theta +\frac{1}{-p}\int _{S}gu^{p}d\theta \) has a minimizer in the class of nonnegative T-periodic functions \(u\in H^{1}(S)\) for which \(\int _{S}u^{2}+u'^{2}-gu^{p}d\theta \ge 0\). This minimizer is used to prove the existence of a positive periodic solution \(u\in H^{1}(S)\) of a one dimensional conformal curvature problem \(-u''+u=\frac{g(\theta )}{u^{1-p}}\).