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Normalized solutions for Schrödinger equations with potentials and general nonlinearities

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Abstract

In this paper, we are concerned with the nonlinear Schrödinger equation

$$\begin{aligned} -\Delta u+V(x)u+\lambda u=g(u)\text { in }{\mathbb {R}}^{N}\text {, }\lambda \in {\mathbb {R}}, \end{aligned}$$

with prescribed \(L^{2}\)-norm \(\int _{{\mathbb {R}}^{N}}u^{2}dx=\rho ^{2}\) and \( \lim _{|x|\rightarrow +\infty }V(x)=:V_{\infty }\le +\infty \) under general assumptions on g which allows at least mass critical growth. For the case of \(V_{\infty }<\infty \), including singular potential, the sufficient conditions are given for the existence of a ground state solution by developing the minimization methods with constraints proposed in Bieganowski and Mederski (J Funct Anal 280(11):108989, 2021) and a delicate analysis of estimates on the least energy comparing with the limiting functional. While for the trapping case \(V_{\infty }=\infty \), the existence of a ground state solution as well as a second solution of mountain pass type is established.

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Acknowledgements

We are grateful to the anonymous referees for valuable comments and suggestions, which helped us to improve our manuscript. This research was supported by the National Natural Science Foundation of China (Nos. 12101020, 12171014, 12171326). We would like to express our gratitude to Prof. J. Schino about some discussions related to Lemma 2.13 and pointing out to us the reference [14].

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Correspondence to Leiga Zhao.

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Communicated by A. Mondino.

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Liu, Y., Zhao, L. Normalized solutions for Schrödinger equations with potentials and general nonlinearities. Calc. Var. 63, 99 (2024). https://doi.org/10.1007/s00526-024-02699-4

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