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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.

    Citation:

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    [6] C. Ji and A. Szulkin, A logarithmic Schrödinger equation with asymptotic conditions on the potential, J. Math. Anal. Appl., 437 (2016), 241-254.  doi: 10.1016/j.jmaa.2015.11.071.
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    [9] M. Squassina and A. Szulkin, Multiple solution to logarithmic Schrödinger equations with periodic potential, Cal. Var. Partial Differential Equations, 54 (2015), 585-597.  doi: 10.1007/s00526-014-0796-8.
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    [12] K. Tanaka and C. X. Zhang, Multi-bump solutions for logarithmic Schrödinger equations, Cal. Var. Partial Differential Equations, 56 (2017), Art. 33, 35 pp. doi: 10.1007/s00526-017-1122-z.
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