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Normalized solutions for a fourth-order Schrödinger equation with a positive second-order dispersion coefficient

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Abstract

In this paper, we study normalized solutions to a fourth-order Schrödinger equation with a positive second-order dispersion coefficient in the mass supercritical regime. Unlike the well-studied case where the second-order term is zero or negative, the geometrical structure of the corresponding energy functional changes dramatically and this makes the solution set richer. Under suitable control of the second-order dispersion coefficient and mass, we find at least two radial normalized solutions, a ground state and an excited state, together with some asymptotic properties. It is worth pointing out that in the considered repulsive case, the compactness analysis of the related Palais-Smale sequences becomes more challenging. This forces the implementation of refined estimates of the Lagrange multiplier and the energy level to obtain normalized solutions.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11901147) and the Fundamental Research Funds for the Central Universities of China (Grant No. JZ2020HGTB0030). The authors are very grateful to Professor Louis Jeanjean whose comments on the first version of this work have permitted us to improve our manuscript and to avoid including a wrong proof in excluding vanishing of local minimizing sequences. The authors thank Professor Gongbao Li for helpful discussions on the whole paper and constant encouragement over the past few years.

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Authors and Affiliations

  1. School of Mathematics, Hefei University of Technology, Hefei, 230009, China

    Xiao Luo

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Tao Yang

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  1. Xiao Luo
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  2. Tao Yang
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Correspondence to Tao Yang.

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Luo, X., Yang, T. Normalized solutions for a fourth-order Schrödinger equation with a positive second-order dispersion coefficient. Sci. China Math. 66, 1237–1262 (2023). https://doi.org/10.1007/s11425-022-1997-3

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  • DOI: https://doi.org/10.1007/s11425-022-1997-3

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