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

Abstract

We consider the following prescribed scalar curvature problem on $S^N$ (*)$$\{\begin{array}{l}-{\Delta }_{S^N}u+\frac{N(N-2)}{2}u=\tilde{K}{u}^{\frac{N+2}{N-2}}on{S}^N,u>0\end{array}$$ 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.

arXiv comment40 pages