Positive normalized solutions of Schrödinger equations with Sobolev critical growth in bounded domains
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916733095247872 |
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| author | Chang, Xiaojun Liu, Manting Yan, Duokui |
| author_facet | Chang, Xiaojun Liu, Manting Yan, Duokui |
| contents | This paper investigates the existence of positive normalized solutions to the Sobolev critical Schrödinger equation: \begin{equation*} \left\{ \begin{aligned} &-Δu +λu =|u|^{2^*-2}u \quad &\mbox{in}& \ Ω,\\ &\int_Ω|u|^{2}dx=c, \quad u=0 \quad &\mbox{on}& \ \partialΩ, \end{aligned} \right. \end{equation*} where $Ω\subset\mathbb{R}^{N}$ ($N\geq3$) is a bounded smooth domain, $2^*=\frac{2N}{N-2}$, $λ\in \mathbb{R}$ is a Lagrange multiplier, and $c>0$ is a prescribed constant. By introducing a novel blow-up analysis for Sobolev subcritical approximation solutions with uniformly bounded Morse index and fixed mass, we establish the existence of mountain pass type positive normalized solutions for
$N\ge 3$. This resolves an open problem posed in [Pierotti, Verzini and Yu, SIAM J. Math. Anal. 2025]. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_07578 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Positive normalized solutions of Schrödinger equations with Sobolev critical growth in bounded domains Chang, Xiaojun Liu, Manting Yan, Duokui Analysis of PDEs 35B33, 35J20, 35J60, 35Q55 This paper investigates the existence of positive normalized solutions to the Sobolev critical Schrödinger equation: \begin{equation*} \left\{ \begin{aligned} &-Δu +λu =|u|^{2^*-2}u \quad &\mbox{in}& \ Ω,\\ &\int_Ω|u|^{2}dx=c, \quad u=0 \quad &\mbox{on}& \ \partialΩ, \end{aligned} \right. \end{equation*} where $Ω\subset\mathbb{R}^{N}$ ($N\geq3$) is a bounded smooth domain, $2^*=\frac{2N}{N-2}$, $λ\in \mathbb{R}$ is a Lagrange multiplier, and $c>0$ is a prescribed constant. By introducing a novel blow-up analysis for Sobolev subcritical approximation solutions with uniformly bounded Morse index and fixed mass, we establish the existence of mountain pass type positive normalized solutions for $N\ge 3$. This resolves an open problem posed in [Pierotti, Verzini and Yu, SIAM J. Math. Anal. 2025]. |
| title | Positive normalized solutions of Schrödinger equations with Sobolev critical growth in bounded domains |
| topic | Analysis of PDEs 35B33, 35J20, 35J60, 35Q55 |
| url | https://arxiv.org/abs/2505.07578 |