Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities
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| Format: | Preprint |
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2025
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| _version_ | 1866917136573661184 |
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| author | Chang, Xiaojun Li, Cong-Mei |
| author_facet | Chang, Xiaojun Li, Cong-Mei |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_09437 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities Chang, Xiaojun Li, Cong-Mei Analysis of PDEs Mathematical Physics 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. |
| title | Normalized solutions of $L^2$ supercritical NLS equations in exterior domains with inhomogeneous nonlinearities |
| topic | Analysis of PDEs Mathematical Physics |
| url | https://arxiv.org/abs/2512.09437 |