Complex group actions on the sphere and sign changing solutions for the CR-Yamabe equation

In this paper we prove that the CR-Yamabe equation on the sphere has infinitely many sign changing solutions. The problem is variational but the related functional does not satisfy the Palais-Smale condition, therefore the standard topological methods fail to apply directly. To overcome this lack of compactness, we will exploit different group actions on the sphere in order to find suitable closed subspaces, on which the restricted functional is Palais-Smale: this will allow us to use the minimax argument of Ambrosetti-Rabinowitz to find critical points. By a classification of the positive solutions and by considerations on the energy blow-up, we will get the desired result.