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Existence and concentration of positive solutions for a logarithmic Schrödinger equation via penalization method

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Abstract

In this article we are concerned with the following logarithmic Schrödinger equation

$$\begin{aligned} \left\{ \begin{array}{lc} -{\epsilon }^2\Delta u+ V(x)u=u \log u^2, &{}\quad \text{ in } \,\, {\mathbb {R}}^{N}, \\ u \in H^1({\mathbb {R}}^{N}), &{} \; \\ \end{array} \right. \end{aligned}$$

where \(\epsilon >0, N \ge 1\) and \(V:{\mathbb {R}}^{N}\rightarrow {\mathbb {R}}\) is a continuous potential. Under a local assumption on the potential V, we use the variational methods to prove the existence and concentration of positive solutions for the above problem.

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Acknowledgements

The authors would like to thank the anonymous referees for their valuable suggestions and comments.

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Authors and Affiliations

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

    Claudianor O. Alves

  2. Department of Mathematics, East China University of Science and Technology, Shanghai, 200237, People’s Republic of China

    Chao Ji

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

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Communicated by P. Rabinowitz.

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Claudianor O. Alves was partially supported by CNPq/Brazil 304804/2017-7. Chao Ji was partially supported by Shanghai Natural Science Foundation (18ZR1409100).

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Alves, C.O., Ji, C. Existence and concentration of positive solutions for a logarithmic Schrödinger equation via penalization method. Calc. Var. 59, 21 (2020). https://doi.org/10.1007/s00526-019-1674-1

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