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Normalized solutions to planar Schrödinger equation with exponential critical nonlinearity

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Abstract

This paper is concerned with the following planar Schrödinger equation

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+\lambda u = f(u),&x \in {\mathbb {R}}^{2},\\&\mathop \int \limits _{{\mathbb {R}}^2}u^2dx=c,&\lambda \in {\mathbb {R}}^+. \end{aligned}\right. \end{aligned}$$

where \(f \in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\) is of critical exponential growth. We obtain the existence of ground state normalized solutions \((u,\lambda )\) under general assumptions, and here \(\lambda \) stands for a Lagrange multiplier. Our theorems extend the results of Alves, Ji and Miyagaki (Calc Var 61:18, 2022) and Chang, Liu and Yan (J Geom Anal 33:83, 2023), where f satisfies a strong global assumption. In particular, some new estimates and approaches are introduced to overcome the lack of compactness resulting from the critical growth of f(u).

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions and comments. The second author wants to thank Chern Institute of Mathematics Visiting Scholar Program.

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Authors and Affiliations

  1. School of Mathematical Sciences and LPMC, Nankai University, Tianjin, 300071, China

    Shuai Mo

  2. Tianjin Chengjian University, 26 Jinjing Road, Tianjin, 300384, China

    Lixia Wang

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  1. Shuai Mo
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L. X.Wang is partially supported by National Natural Science Foundation of China (11801400) and (11571187) and Scientific Research Program of Tianjin Education Commission (2020KJ045 and 2023KJ216).

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Mo, S., Wang, L. Normalized solutions to planar Schrödinger equation with exponential critical nonlinearity. Z. Angew. Math. Phys. 75, 26 (2024). https://doi.org/10.1007/s00033-023-02158-8

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