Nondegeneracy of Nodal Solutions to the Critical Yamabe Problem

Abstract

We prove the existence of a sequence of nondegenerate, in the sense of Duyckaerts–Kenig–Merle [9], nodal nonradial solutions to the critical Yamabe problem $$-\Delta Q= |Q|^{\frac{2}{n-2}} Q, \quad Q \in {\mathcal D}^{1,2}(\mathbb{R}^n).$$ This is the first example in the literature of nondegeneracy for nodal nonradial solutions of nonlinear elliptic equations and it is also the only nontrivial example for which the result of Duyckaerts–Kenig–Merle [9] applies.