Let \(\lambda _2\) denote the second eigenvalue of the fixed membrane problem in a bounded domain \(\Omega \subseteq \mathbb {R}^N\) with \(N\ge 2\) . Use \(\Lambda _1\) to denote the first eigenvalue of the buckling problem with the same domain as the fixed membrane problem. We show that \(\Lambda _1\ge \lambda _2\) with the equality holds if and only if \(\Omega \) is a ball. For \(N=2\) , this conclusion is the famous Weinstein conjecture, which was solved by Payne in 1955. As one of its applications, we consider the overdetermined problem \(\begin{aligned} \left\{ \begin{array}{ll} \Delta u+\alpha u=0\,\, & \text {in }\Omega ,\\ \partial _\nu u=0\,\, & \text {on }\partial \Omega ,\\ u=c\,\, & \text {on }\partial \Omega , \end{array}\right. \end{aligned}\) and give a confirmed answer to the Schiffer conjecture: when \(\alpha =\lambda _2\) , \(\Omega \) must be a ball. This extends the corresponding planar conclusion due to Berenstein in 1980.
