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Semiclassical States for Coupled Nonlinear Schrödinger System with Competing Potentials

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Abstract

In this paper, we study the following coupled nonlinear Schrödinger system with potentials

$$\begin{aligned} \left\{ \begin{array}{ll} -\epsilon ^{2}\Delta u +u+V(x)v=W(x)f(|z|)v~~\hbox {in}~\mathbb {R}^{N},\\ -\epsilon ^{2}\Delta v +v+V(x)u=W(x)f(|z|)u~~\hbox {in}~\mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$

where \(z=(u,v):\mathbb {R}^{N}\rightarrow \mathbb {R}^{2}\), \(N\ge 3\), and \(\epsilon \) is a small positive parameter. We assume that the potentials V, W, and nonlinearity f are continuous but are not necessarily of class \(C^{1}\). Combining this with other suitable conditions on f, we prove the existence of semiclassical ground state solutions via generalized Nehari manifold method. Moreover, we determine two concrete sets related to the potentials V and W as the concentration positions and we describe the concentration of these ground state solutions as \(\epsilon \rightarrow 0\). The results presented in this paper improve and extend the previous relevant results.

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Acknowledgements

This work was supported by the Natural Science Foundation of Hunan Province (2021JJ30189), the Key project of Scientific Research Project of Department of Education of Hunan Province (21A0387), the China Scholarship Council (Nos. 201908430218, 201908430219) for visiting the University of Craiova (Romania), and Funding scheme for Young Backbone Teachers of universities in Hunan Province (Hunan Education Notification (2018) No. 574 and (2020) No. 43). The authors would like to thank the China Scholarship Council and the Embassy of the People’s Republic of China in Romania.

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Authors and Affiliations

  1. College of Science, Hunan University of Technology and Business, Changsha, 410205, Hunan, People’s Republic of China

    Jian Zhang & Wen Zhang

  2. Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Hunan University of Technology and Business, Changsha, 410205, Hunan, People’s Republic of China

    Jian Zhang & Wen Zhang

  3. Department of Mathematics, University of Craiova, 200585, Craiova, Romania

    Jian Zhang & Wen Zhang

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  1. Jian Zhang
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Correspondence to Wen Zhang.

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Zhang, J., Zhang, W. Semiclassical States for Coupled Nonlinear Schrödinger System with Competing Potentials. J Geom Anal 32, 114 (2022). https://doi.org/10.1007/s12220-022-00870-x

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