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Normalized solutions for a Schrödinger equation with critical growth in \({\mathbb {R}}^{N}\)

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Abstract

In this paper we study the existence of normalized solutions to the following nonlinear Schrödinger equation with critical growth

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u={\lambda u}+f(u), &{} \quad \hbox {in }{\mathbb {R}}^N,\\ u>0,&{}\quad \int _{{\mathbb {R}}^{N}}|u|^{2}dx=a^{2}, \end{array} \right. \end{aligned}$$

where \(a>0\), \(\lambda \in {\mathbb {R}}\) and f has an exponential critical growth when \(N=2\), and \(f(t)=\mu |t|^{q-2}t+|t|^{2^*-2}t\) with \(q \in (2+\frac{4}{N},2^*)\), \(\mu >0\) and \(2^*=\frac{2N}{N-2}\) when \(N \ge 3\). Our main results complement some recent results for \(N \ge 3\) and it is totally new for \(N=2\).

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Acknowledgements

The authors would like to thank the anonymous referee for several valuable suggestions and comments which helped to improve the paper.

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Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP:58429-900, Brazil

    Claudianor O. Alves

  2. School of Mathematics, East China University of Science and Technology, Shanghai, 200237, People’s Republic of China

    Chao Ji

  3. Departamento de Matemática, Universidade Federal de São Carlos, São Carlos, SP, CEP:13565-905, Brazil

    Olimpio H. Miyagaki

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  1. Claudianor O. Alves
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  2. Chao Ji
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  3. Olimpio H. Miyagaki
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Correspondence to Chao Ji.

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Communicated by P. H. Rabinowitz.

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C.O. Alves was partially supported by CNPq/Brazil Grant 304804/2017-7. C. Ji was partially supported by National Natural Science Foundation of China (No. 12171152) and Natural Science Foundation of Shanghai (No. 20ZR1413900). O. H. Miyagaki was supported by FAPESP/Brazil Grant 2019/24901-3 and CNPq/Brazil Grant 307061/2018-3.

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Alves, C.O., Ji, C. & Miyagaki, O.H. Normalized solutions for a Schrödinger equation with critical growth in \({\mathbb {R}}^{N}\). Calc. Var. 61, 18 (2022). https://doi.org/10.1007/s00526-021-02123-1

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