An Observation on Eigenfunctions of the Laplacian

In his seminal 1943 paper F. Rellich proved that, in the complement of a cavity \(\Omega = \{x\in \mathbb {R}^n\mid |x|>R_0\}\) $\mathrm{\Omega }=\{x\in {\mathbb{R}}^n\mid |x|>R_0\}$ , there exist no nontrivial solution f of the Helmholtz equation \(\Delta f = - \lambda f\) $\mathrm{\Delta }f=-\lambda f$ , when \(\lambda >0\) $\lambda >0$ , such that \(\int _{\Omega } |f|^2 dx < \infty \) ${\int }_{\mathrm{\Omega }}{|f|}^{2}dx<\infty$ . In this note we generalise this result by showing that if \(\int _{\Omega } |f|^p dx < \infty \) ${\int }_{\mathrm{\Omega }}{|f|}^{p}dx<\infty$ for some \(0<p\le \frac{2n}{n-1}\) $0<p\le \frac{2n}{n-1}$ , then \(f\equiv 0\) $f\equiv 0$ . This result is sharp since for any \(p> \frac{2n}{n-1}\) $p>\frac{2n}{n-1}$ , eigenfunctions do exist in \(\Omega \) $\mathrm{\Omega }$ .