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Multiple positive solutions for a Schrödinger logarithmic equation

  • * Corresponding author: Chao Ji

    * Corresponding author: Chao Ji

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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  • 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.

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


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