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Multiplicity of Normalized Solutions for Schrödinger Equations

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Abstract

In this paper, we consider the following nonlinear Schrödinger equation with an \(L^2\)-constraint:

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

where \(N\ge 3\), \(a,\mu >0\), \(2<q<2+\frac{4}{N}<p<2^*\), \(2q+2N-pN<0\) and \(\lambda \in \mathbb {R}\) arises as a Lagrange multiplier. We deal with the concave and convex cases of energy functional constraints on the \(L^2\) sphere, and prove the existence of infinitely solutions with positive energy levels.

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Acknowledgements

The authors thanks the editors and reviewers for their valuable comments and suggestions, which greatly improved the manuscript.

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Correspondence to Gui-Dong Li.

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Communicated by Rosihan M. Ali.

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Gui-Dong Li was supported by the special (special post) scientific research fund of natural science of Guizhou University (No.(2021)43), Guizhou Provincial Education Department Project (No.(2022)097), Guizhou Provincial Science and Technology Projects (No.[2023]YB033, [2023]YB036) and National Natural Science Foundation of China (No.12201147).

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Lv, YC., Li, GD. Multiplicity of Normalized Solutions for Schrödinger Equations. Bull. Malays. Math. Sci. Soc. 47, 113 (2024). https://doi.org/10.1007/s40840-024-01713-4

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