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arXiv · 2512.09437

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

Xiaojun Chang·Cong-Mei Li
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Abstract

This paper establishes the existence of normalized mountain pass solutions to the $L^2$-supercritical nonlinear Schrödinger equation with inhomogeneous nonlinearity $|x|^{-α}|u|^{p-2}u$ in exterior domains. In contrast, for the autonomous case ($α=0$), 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 $|x|^{-α}$ 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.

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Xiaojun Chang, Cong-Mei Li. 2025-12-10. Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities. https://arxiv.org/abs/2512.09437

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