Low energy nodal solutions to the Yamabe equation

Abstract

Given an isoparametric function f on the n-dimensional sphere, we consider the space of functions w∘f to reduce the Yamabe equation on the round sphere into a singular ODE on w in the interval [0,π], of the form w″+(h(r)/sin⁡r)w′+λ(|w|4/n−2w−w)=0 and boundary conditions w′(0)=0=w′(π), where h is a monotone function with exactly one zero on (0,π) and λ>0 is a constant. For any positive integer k we obtain a solution with exactly k-zeroes yielding solutions to the Yamabe equation with exactly k connected isoparametric hypersurfaces as nodal set. The idea of the proof is to consider the initial value problems on the singularities, and then to solve the corresponding double shooting problem, matching the values of w and w′ at the unique zero of h. In particular we obtain solutions with exactly one zero, providing solutions of the Yamabe equation with low energy.