Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2024-01-28 , DOI: 10.1007/s00033-023-02158-8 Shuai Mo , Lixia Wang
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).
中文翻译：
具有指数临界非线性的平面薛定谔方程的归一化解
本文关注以下平面薛定谔方程
$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u+\lambda u = f(u),&x \in {\mathbb {R}}^{2},\\&\mathop \int \limits _{{\mathbb {R}}^2}u^2dx=c,&\lambda \in {\mathbb {R}}^+。\end{对齐}\右。\end{对齐}$$其中\(f \in {\mathcal {C}}({\mathbb {R}},{\mathbb {R}})\)具有临界指数增长。我们在一般假设下得到基态归一化解\((u,\lambda )\)的存在性，这里\(\lambda \)代表拉格朗日乘子。我们的定理扩展了 Alves、Ji 和 Miyagaki (Calc Var 61:18, 2022) 以及 Chang、Liu 和 Yan (J Geom Anal 33:83, 2023) 的结果，其中f满足强全局假设。特别是，引入了一些新的估计和方法来克服f ( u )临界增长导致的紧致性不足。
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