Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential

Abstract

In this article we use the variational method developed by Szulkin (Ann Inst H Poincaré Anal Non Linéire 3:77–109, 1986) to prove the existence of a positive solution for the following logarithmic Schrödinger equation

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

where \(\epsilon >0, N \ge 1\) and V is a saddle-like potential.

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Acknowledgements

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

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Correspondence to Chao Ji.

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

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Alves, C.O., Ji, C. Existence of a positive solution for a logarithmic Schrödinger equation with saddle-like potential. manuscripta math. 164, 555–575 (2021). https://doi.org/10.1007/s00229-020-01197-z

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Mathematics Subject Classification

  • 35A15
  • 35J10
  • 35B09