Nonexistence of minimizers for the second conformal eigenvalue near the round sphere in low dimensions

We consider the problem of minimizing the second conformal eigenvalue of the conformal Laplacian in a conformal class of metrics with renormalized volume. We prove, in dimensions $n\in\left\{3,\dotsc,10\right\}$, that a minimizer for this problem does not exist for metrics sufficiently close to the round metric on the sphere. This is in striking contrast with the situation in dimensions $n \ge 11$, where Ammann and Humbert obtained the existence of minimizers for the second conformal eigenvalue on any smooth closed non-locally conformally flat manifold. As a byproduct of our techniques, we also obtain a lower bound on the energy of sign-changing solutions of the \nobreak Yamabe equation in dimensions 3, 4 and 5, which extends a result obtained by Weth in the case of the round sphere.