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

We consider the prescribed scalar curvature problem on $$ {{\mathbb {S}}}^N $$ S N $$\begin{aligned} \Delta _{{{\mathbb {S}}}^N} v-\frac{N(N-2)}{2} v+{\tilde{K}}(y) v^{\frac{N+2}{N-2}}=0 \quad \text{ on } \ {{\mathbb {S}}}^N, \qquad v >0 \quad {\quad \hbox {in } }{{\mathbb {S}}}^N, \end{aligned}$$ Δ S N v - N ( N - 2 ) 2 v + K ~ ( y ) v N + 2 N - 2 = 0 on S N , v > 0 in S N , under the assumptions that the scalar curvature $${\tilde{K}}$$ 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.