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Existence of a positive solution for a class of Schrödinger logarithmic equations on exterior domains

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Abstract

In this paper we will prove the existence of a positive solution for a class of Schrödinger logarithmic equation of the form.

$$\begin{aligned} \left\{ \begin{aligned} -\Delta u&+ u =Q(x)u\log u^2,\;\;\hbox {in}\;\;\Omega , \\&{\mathcal {B}}u=0 \,\,\, \hbox {on} \,\,\, \partial \Omega , \end{aligned} \right. \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^N\), \(N \ge 3\), is an exterior domain, i.e., \(\Omega ^c={\mathbb {R}}^N {\setminus } \Omega \) is a bounded smooth domain where \({\mathcal {B}}u=u\) or \({\mathcal {B}}u=\frac{\partial u}{\partial \nu }\). We have used new approach that allows us to apply the usual \(C^1\)-variational methods to get a nontrivial solutions for these classes of problems.

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Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, 58429-970, Brazil

    Claudianor O. Alves

  2. Centro de Ciências Exatas e Sociais Aplicadas, Universidade Estadual da Paraíba, Patos, PB, 58706-550, Brazil

    Ismael S. da Silva

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  1. Claudianor O. Alves
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  2. Ismael S. da Silva
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Correspondence to Ismael S. da Silva.

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Alves, C.O., da Silva, I.S. Existence of a positive solution for a class of Schrödinger logarithmic equations on exterior domains. Z. Angew. Math. Phys. 75, 77 (2024). https://doi.org/10.1007/s00033-024-02212-z

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