Absence of $L^p$ spectrum for asymptotically flat diffusions in region with cavities

We study solutions to variable-coefficient elliptic equations of the form $-\text{D}(A(x)\mathrm{\nabla }u)=\kappa u$, $\kappa >0$, in an exterior domain $\text{Om}\subset \text{Rn}$, where $A(x)$ is uniformly elliptic and asymptotically flat. Extending Rellich’s classical $L^2$ result for the Laplacian, we show that if $u\in L^p(\text{Om})$ for some $0<p<\frac{2n}{n-1}$, then $u\equiv 0$. The proof uses new monotonicity formulas based on weighted energies and vector fields adapted to the geometry of $A(x)$. Our results highlight a sharper integrability threshold in the variable-coefficient setting.