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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 rotationallysymmetric, and has a positive local maximum point between the poles. We provethe existence of infinitely many non-radial positive solutions, whose energycan be made arbitrarily large. These solutions are invariant under somenon-trivial sub-group of $O(3)$ obtained doubling the equatorial. We use thefinite dimensional Lyapunov-Schmidt reduction method.

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