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Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well

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Abstract

This article concerns the existence of multi-bump positive solutions for the following logarithmic Schrödinger equation:

$$\left\{ {\begin{array}{*{20}{l}} { - \Delta u + \lambda V(x)u = u\log {u^2}\;\;\;in\;\;\;{\mathbb{R}^N},} \\ {u \;\in {H^1}({R^N}),} \end{array}} \right.$$

where N ⩾ 1, ⋋ > 0 is a parameter and the nonnegative continuous function V: ℝN → ℝ has potential well Ω:= int V−1(0) which possesses k disjoint bounded components \({\rm{\Omega}}\,{\rm{=}}\, \cup _{j = 1}^k{{\rm{\Omega}}_j}\). Using the variational methods, we prove that if the parameter ⋋ > 0 is large enough, then the equation has at least 2k − 1 multi-bump positive solutions.

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Acknowledgements

The first author was supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico-CNPq/Brazil (Grant No. 304804/2017-7). The second author was supported by Natural Science Foundation of Shanghai (Grant Nos. 20ZR1413900 and 18ZR1409100). The authors thank the referees for many helpful comments which clarify the paper.

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Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, 58429-900, Brazil

    Claudianor O. Alves

  2. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, China

    Chao Ji

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  1. Claudianor O. Alves
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  2. Chao Ji
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Correspondence to Claudianor O. Alves or Chao Ji.

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Alves, C.O., Ji, C. Multi-bump positive solutions for a logarithmic Schrödinger equation with deepening potential well. Sci. China Math. 65, 1577–1598 (2022). https://doi.org/10.1007/s11425-020-1821-9

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  • DOI: https://doi.org/10.1007/s11425-020-1821-9

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