Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities
Xiaojun Chang, Cong-Mei Li
TL;DR
This work demonstrates the existence of normalized mountain-pass solutions for an L^2–supercritical NLS with an inhomogeneous, spatially decaying nonlinearity in exterior domains. By breaking the scaling symmetry through the term |x|^{−α} with α>0, energy leakage to infinity is suppressed, allowing a separation of exterior-domain and whole-space energy levels. A novel combination of a monotonicity trick, constrained Morse-index controls, and careful blow-up analysis yields a positive, small-mass normalized solution with λ>0. The results hinge on a uniform mountain-pass structure for a family of constrained functionals and a detailed asymptotic/compactness analysis showing convergence to localized ground-state profiles. This provides a rigorous mechanism for trap-like behavior in non-autonomous nonlinearities on non-compact domains, with potential relevance to nonlinear optics and Bose-Einstein condensates.
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.