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

Abstract

AbstractWe consider the prescribed scalar curvature problem on$$ {{\mathbb {S}}}^N $$SN$$\begin{aligned} \Delta _{{{\mathbb {S}}}^N} v-\frac{N(N-2)}{2} v+{\tilde{K}}(y) v^{\frac{N+2}{N-2}}=0 \quad \text{ on } \ {{\mathbb {S}}}^N, \qquad v >0 \quad {\quad \hbox {in } }{{\mathbb {S}}}^N, \end{aligned}$$ΔSNv-N(N-2)2v+K~(y)vN+2N-2=0onSN,v>0inSN,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 ofO(3) obtained doubling the equatorial. We use the finite dimensional Lyapunov–Schmidt reduction method.