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Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems

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Abstract

This paper is dedicated to studying the following elliptic system of Hamiltonian type:

$$\begin{cases}-\varepsilon^2\Delta{u}+u+V(x)v=Q(x)F_v{(u,v)}, & x \in \mathbb{R}^N,\\-\varepsilon^2\Delta{v}+v+V(x)u=Q(x)F_u{(u,v)}, & x \in \mathbb{R}^N,\\|u(x)|+|v(x)|\rightarrow0, & as |x|\rightarrow \infty\end{cases}$$

where N ⩾ 3, V, \(Q\in\mathcal{C}(\mathbb{R}^N,\mathbb{R})\), V (x) is allowed to be sign-changing and inf Q > 0, and \(F\in\mathcal{C}^1(\mathbb{R}^2,\mathbb{R})\) is superquadratic at both 0 and infinity but subcritical. Instead of the reduction approach used in Ding et al. (2014), we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al. (2014). We can find an ε0 > 0 which is determined by terms of N, V, Q and F, and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for all ε ∈ (0, ε0].

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11171351).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Central South University, Changsha, 410083, China

    Xianhua Tang

  2. Department of Mathematics, Huaihua University, Huaihua, 418008, China

    Xiaoyan Lin

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  1. Xianhua Tang
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  2. Xiaoyan Lin
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Correspondence to Xianhua Tang.

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Tang, X., Lin, X. Existence of ground state solutions of Nehari-Pankov type to Schrödinger systems. Sci. China Math. 63, 113–134 (2020). https://doi.org/10.1007/s11425-017-9332-3

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  • DOI: https://doi.org/10.1007/s11425-017-9332-3

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