Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities

Abstract

This paper establishes the existence of normalized mountain pass solutions to the $L^2$-supercritical nonlinear Schr\"odinger equation with inhomogeneous nonlinearity $|x|^{-\alpha}|u|^{p-2}u$ in exterior domains. In contrast, for the autonomous case ($\alpha=0$), Appolloni \& Molle (2025) and Zhang \& Zhang (2022) showed that potential mountain pass solutions share the same energy levels as in $\mathbb{R}^N$, causing non-existence due to energy leakage to infinity. This work demonstrates that the physically motivated decaying term $|x|^{-\alpha}$ breaks the scaling symmetry inherent in the autonomous case. Such breaking energetically separates the exterior domain problem from the whole space one and thereby prevents energy leakage. Using a novel min-max argument that combines monotonicity trick, Morse index estimates, and blow-up analysis, we prove the existence of a positive mountain pass solution for sufficiently small mass, revealing a new phenomenon of non-autonomous nonlinearities in non-compact domains.