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On existence and concentration of solutions to a class of quasilinear problems involving the 1-Laplace operator

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Abstract

In this work we use variational methods to prove results on existence and concentration of solutions to a problem in \(\mathbb {R}^N\) involving the 1-Laplacian operator. A thorough analysis on the energy functional defined in the space of functions of bounded variation \(BV(\mathbb {R}^N)\) is necessary, where the lack of compactness is overcome by using the Concentration of Compactness Principle due to Lions.

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Acknowledgements

M.T.O. Pimenta has been supported by FAPESP 2017/01756-2 and CNPq/Brazil 442520/2014-0. C.O. Alves was partially supported by CNPq/Brazil 304036/2013-7 and INCT-MAT. The authors would like to thank to the referee for his/her important comments.

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Authors and Affiliations

  1. Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, Campina Grande, PB, 58429-900, Brazil

    Claudianor O. Alves

  2. Departamento de Matemática e Computação, Faculdade de Ciências e Tecnologia, Universidade Estadual Paulista (Unesp), Presidente Prudente, SP, 19060-900, Brazil

    Marcos T. O. Pimenta

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  1. Claudianor O. Alves
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  2. Marcos T. O. Pimenta
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Correspondence to Marcos T. O. Pimenta.

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Communicated by P. Rabinowitz.

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Alves, C.O., Pimenta, M.T.O. On existence and concentration of solutions to a class of quasilinear problems involving the 1-Laplace operator. Calc. Var. 56, 143 (2017). https://doi.org/10.1007/s00526-017-1236-3

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