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On existence of multiple normalized solutions to a class of elliptic problems in whole \({\mathbb {R}}^N\)

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Abstract

In this paper, we study the existence of multiple normalized solutions to the following class of elliptic problems

$$\begin{aligned} \left\{ \begin{aligned}&-\Delta u=\lambda u+h(\epsilon x)f(u), \quad \hbox {in}\; {\mathbb {R}}^N,\\&\int \limits _{{\mathbb {R}}^{N}}|u|^{2}\hbox {d}x=a^{2}, \end{aligned} \right. \end{aligned}$$

where \(a,\epsilon >0\), \(\lambda \in {\mathbb {R}}\) is an unknown parameter that appears as a Lagrange multiplier, \(h:{\mathbb {R}}^N \rightarrow [0,\infty )\) is a continuous function, and f is continuous function with \(L^2\)-subcritical growth. It is proved that the numbers of normalized solutions are at least the numbers of global maximum points of h when \(\epsilon \) is small enough.

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Acknowledgements

The author thanks Prof. Louis Jeanjean for useful remarks on a preliminary version of this work.

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  1. Unidade Acadêmica de Matemática Universidade Federal de Campina Grande, Campina Grande, PB, 58429-970, Brazil

    Claudianor O. Alves

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Correspondence to Claudianor O. Alves.

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C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7.

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Alves, C.O. On existence of multiple normalized solutions to a class of elliptic problems in whole \({\mathbb {R}}^N\). Z. Angew. Math. Phys. 73, 97 (2022). https://doi.org/10.1007/s00033-022-01741-9

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