Desingularization of Clifford Torus and Nonradial Solutions to Yamabe Problem with Maximal Rank

Abstract

Through desingularization of Clifford torus, we prove the existence of a sequence of nondegenerate (in the sense of Duyckaerts-Kenig-Merle nodal nonradial solutions to the critical Yamabe problem $$-\Delta u=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u,\qquad u\in {\mathcal{D}}^{1,2}(\mathcal{R}^n). $$ The case $n=4$ is the first example in the literature of a solution with {\em maximal rank} ${\mathcal N}=2n+1+\frac{n(n-1)}{2}$.