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

Abstract

We consider the prescribed scalar curvature problem on {{mathbb {S}}}^N ΔSNv-N(N-2)2v+K~(y)vN+2N-2=0onSN,v>0inSN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\Delta _{{{\\mathbb {S}}}^N} v-\\frac{N(N-2)}{2} v+{\ ilde{K}}(y) v^{\\frac{N+2}{N-2}}=0 \\quad \ ext{ on } \\ {{\\mathbb {S}}}^N, \\qquad v >0 \\quad {\\quad \\hbox {in } }{{\\mathbb {S}}}^N, \\end{aligned}$$\\end{document}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.