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

TL;DR

This paper proves the existence of normalized solutions for a supercritical nonlinear Schrödinger equation with inhomogeneous nonlinearity in exterior domains, highlighting how the inhomogeneity prevents energy leakage and differs from autonomous cases.

Contribution

It introduces a novel min-max approach to establish solutions in non-compact domains with inhomogeneous nonlinearities, breaking the symmetry seen in autonomous cases.

Findings

Existence of positive mountain pass solutions for small mass.

Inhomogeneity prevents energy leakage to infinity.

Breaks scaling symmetry in exterior domain problems.

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.