Normalized solutions of L² supercritical NLS equations in exterior domains with inhomogeneous nonlinearities

2025/12/10 by Chang, Xiaojun, Li, Cong-Mei
#Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2512.09437

Abstract

This paper establishes the existence of normalized mountain pass solutions to the L²-supercritical nonlinear Schrödinger equation with inhomogeneous nonlinearity |x|^-α|u|^p-2u 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 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.

Citations

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Related

• Normalized solutions of L²-supercritical NLS equations on noncompact metric graphs
• Solutions with prescribed mass for L²-supercritical NLS equations under Neumann boundary conditions
• Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations
• Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains
• Normalized solutions of L²-supercritical NLS equations on compact metric graphs