Infinitely many solution for prescribed curvature problem on $S^N$

TL;DR

This paper proves the existence of infinitely many non-radial positive solutions to a prescribed scalar curvature problem on the sphere, under certain symmetry and maximum point conditions on the prescribed function.

Contribution

It establishes the existence of infinitely many solutions with arbitrarily large energy for a class of prescribed curvature equations on the sphere, extending previous results.

Findings

Existence of infinitely many solutions when the prescribed function has a local maximum.

Solutions can have arbitrarily large energy.

Solutions are non-radial and positive.

Abstract

We consider the following prescribed scalar curvature problem on $ S^N$ (*)$$\left\{\begin{array}{l} - \Delta_{S^N} u + \frac{N(N-2)}{2} u = \tilde{K} u^{\frac{N+2}{N-2}} {on} S^N, u >0 \end{array}\right. $$ where $ \tilde{K}$ is positive and rotationally symmetric. We show that if $\tilde{K}$ has a local maximum point between the poles then equation (*) has {\bf infinitely many non-radial positive} solutions, whose energy can be made arbitrarily large.