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Localization of normalized solutions for saturable nonlinear Schrödinger equations

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Abstract

In this paper, we study the existence and concentration behavior of the semiclassical states with L2-constraints for the following saturable nonlinear Schrödinger equation:

$$-\varepsilon^{2}\Delta v+\Gamma{{I(x)+v^{2}}\over{1+I(x)+v^{2}}}v=\lambda v\;\;\;\;\;\text{for}\;x\in\mathbb{R}^{2}.$$

For a negatively large coupling constant Γ, we show that there exists a family of normalized positive solutions (i.e., with the L2-constraint) when ε is small, which concentrate around local maxima of the intensity function I(x) as ε → 0. We also consider the case where I(x) may tend to —1 at infinity and the existence of multiple solutions. The proof of our results is variational and the novelty of the work lies in the development of a new truncation-type method for the construction of the desired solutions.

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Acknowledgements

Xiaoming Wang was supported by National Natural Science Foundation of China (Grant No. 11861053). Zhi-Qiang Wang was supported by National Natural Science Foundation of China (Grant No. 11831009). Xu Zhang was supported by National Natural Science Foundation of China (Grant No. 11901582). The authors thank the referees for helpful comments.

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Authors and Affiliations

  1. School of Mathematics & Computer Science, Shangrao Normal University, Shangrao, 334001, China

    Xiaoming Wang

  2. College of Mathematics and Statistics, Fujian Normal University, Fuzhou, 350117, China

    Zhi-Qiang Wang

  3. Department of Mathematics and Statistics, Utah State University, Logan, UT, 84322, USA

    Zhi-Qiang Wang

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

    Xu Zhang

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  1. Xiaoming Wang
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  2. Zhi-Qiang Wang
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Correspondence to Xu Zhang.

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Wang, X., Wang, ZQ. & Zhang, X. Localization of normalized solutions for saturable nonlinear Schrödinger equations. Sci. China Math. 66, 2495–2522 (2023). https://doi.org/10.1007/s11425-022-2052-1

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