Multiple positive solutions for a Schrödinger logarithmic equation

Claudianor O. Alves; Chao Ji

doi:10.3934/dcds.2020145

Article Contents

2020, Volume 40, Issue 5: 2671-2685. Doi: 10.3934/dcds.2020145

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Multiple positive solutions for a Schrödinger logarithmic equation

Claudianor O. Alves^1, and
Chao Ji^2, ,

1.
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, CEP:58429-900, Brazil
2.
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, China

^* Corresponding author: Chao Ji
^* Corresponding author: Chao Ji

Received: April 30, 2019

Revised: November 30, 2019

Published: March 2020

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

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Abstract

This article concerns with the existence of multiple positive solutions for the following logarithmic Schrödinger equation

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

where $ \epsilon >0 $, $ N \geq 1 $ and $ V $ is a continuous function with a global minimum. Using variational method, we prove that for small enough $ \epsilon>0 $, the "shape" of the graph of the function $ V $ affects the number of nontrivial solutions.

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Mathematics Subject Classification: Primary: 35A15, 35J10; Secondary: 35B09.

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References

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