Doubling the equatorial for the prescribed scalar curvature problem on ${\mathbb{S}}^N$

TL;DR

This paper proves the existence of infinitely many non-radial solutions to a scalar curvature problem on the sphere, using symmetry and reduction methods, with solutions having arbitrarily large energy.

Contribution

It introduces a novel approach to construct multiple non-radial solutions by doubling the equatorial symmetry on the sphere.

Findings

Existence of infinitely many solutions with large energy.

Solutions are invariant under a subgroup of symmetries.

Method employs Lyapunov-Schmidt reduction.

Abstract

We consider the prescribed scalar curvature problem on $ {\mathbb{S}}^N $ $$ \Delta_{{\mathbb S}^N} v-\frac{N(N-2)}{2} v+\tilde{K}(y) v^{\frac{N+2}{N-2}}=0 \quad \mbox{on} \ {\mathbb S}^N, \qquad v >0 \quad \mbox{on} \ {\mathbb S}^N, $$ under the assumptions that the scalar curvature $\tilde K$ is rotationally symmetric, and has a positive local maximum point between the poles. We prove the existence of infinitely many non-radial positive solutions, whose energy can be made arbitrarily large. These solutions are invariant under some non-trivial sub-group of $O(3)$ obtained doubling the equatorial. We use the finite dimensional Lyapunov-Schmidt reduction method.