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Non-Nehari Manifold Method for Hamiltonian Elliptic System with Hardy Potential: Existence and Asymptotic Properties of Ground State Solution

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Abstract

This paper is dedicated to studying ground state solution for a class of Hamiltonian elliptic system with gradient term and inverse square potential. The resulting problem engages four major difficulties: one is that the associated functional is strongly indefinite, the second is that, due to the asymptotically periodic assumption, the associated functional loses the \(\mathbb {Z}^N\)-translation invariance. The third difficulty we must overcome lies in verifying the link geometry and showing the boundedness of Cerami sequences when the non-linearity is asymptotically quadratic. The last is singular potential \(\frac{\mu }{|x|^2}\), which does not belong to the Kato’s class. These enable us to develop a direct approach and new tricks to overcome the difficulties caused by singularity and the dropping of periodicity of potential. We establish the existence and non-existence results of ground state solutions under some mild conditions, and derive asymptotical behavior of ground state solutions.

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Acknowledgements

This research by the authors were supported by the State Scholarship Fund organized by the China Scholarship Council (CSC). This paper was completed when the first author was visiting the Department of Mathematics, University of British Columbia. They would also like to thank Professor Juncheng Wei for his hospitality and helpful discussions and suggestions.

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Authors and Affiliations

  1. Three Gorges Mathematical Research Center, China Three Gorges University, Yichang, 443002, Hubei, People’s Republic of China

    Peng Chen

  2. College of Science, China Three Gorges University, Yichang, 443002, Hubei, People’s Republic of China

    Peng Chen

  3. School of Mathematics and Statistics, Central South University, Changsha, 410083, Hunan, People’s Republic of China

    Xianhua Tang & Limin Zhang

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  1. Peng Chen
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  2. Xianhua Tang
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  3. Limin Zhang
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Correspondence to Peng Chen.

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This work is partially supported by Natural Science Foundation of Hubei Province of China (No. 2021CFB473) and Hubei Educational Committee (No. D20161206)

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Chen, P., Tang, X. & Zhang, L. Non-Nehari Manifold Method for Hamiltonian Elliptic System with Hardy Potential: Existence and Asymptotic Properties of Ground State Solution. J Geom Anal 32, 46 (2022). https://doi.org/10.1007/s12220-021-00739-5

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  • DOI: https://doi.org/10.1007/s12220-021-00739-5

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