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Normalized Ground State Solutions of Nonlinear Schrödinger Equations Involving Exponential Critical Growth

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Abstract

We are concerned with the following nonlinear Schrödinger equation:

$$\begin{aligned} \begin{aligned} {\left\{ \begin{array}{ll} -\Delta u+\lambda u=f(u) \ \ \textrm{in}\ \mathbb {R}^{2},\\ u\in H^{1}(\mathbb {R}^{2}),~~~ \int _{\mathbb {R}^2}u^2dx=\rho , \end{array}\right. } \end{aligned} \end{aligned}$$

where \(\rho >0\) is given, \(\lambda \in \mathbb {R}\) arises as a Lagrange multiplier and f satisfies an exponential critical growth. Without assuming the Ambrosetti–Rabinowitz condition, we show the existence of normalized ground state solutions for any \(\rho >0\). The proof is based on a constrained minimization method and the Trudinger–Moser inequality in \(\mathbb {R}^2\).

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Acknowledgements

The research of Xiaojun Chang is supported by National Natural Science Foundation of China (Grant No. 11971095), while Duokui Yan is supported by National Natural Science Foundation of China (Grant No. 11871086). This work was done when Xiaojun Chang visited the Laboratoire de Mathématiques, Université de Bourgogne Franche-Comté during the period from 2021 to 2022 under the support of China Scholarship Council (Grant No. 202006625034), and he would like to thank the Laboratoire for their support and kind hospitality.

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  1. Manting Liu and Duokui Yan have contributed equally to this work.

Authors and Affiliations

  1. School of Mathematics and Statistics & Center for Mathematics and Interdisciplinary Sciences, Northeast Normal University, Changchun, 130024, China

    Xiaojun Chang & Manting Liu

  2. School of Mathematical Sciences, Beihang University, Beijing, 100191, China

    Duokui Yan

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  1. Xiaojun Chang
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  2. Manting Liu
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Correspondence to Xiaojun Chang.

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Chang, X., Liu, M. & Yan, D. Normalized Ground State Solutions of Nonlinear Schrödinger Equations Involving Exponential Critical Growth. J Geom Anal 33, 83 (2023). https://doi.org/10.1007/s12220-022-01130-8

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