Milan Journal of Mathematics

, Volume 86, Issue 1, pp 15–30 | Cite as

Strauss’ and Lions’ Type Results in \({BV (\mathbb{R}^N}\)) with an Application to an 1-Laplacian Problem

  • Giovany M. Figueiredo
  • Marcos T. O. Pimenta


In this work we state and prove versions of some classical results, in the framework of functionals defined in the space of functions of bounded variation in \({\mathbb{R}^N}\). More precisely, we present versions of the Radial Lemma of Strauss, the compactness of the embeddings of the space of radially symmetric functions of BV (\({\mathbb{R}^N}\)) in some Lebesgue spaces and also a version of the Lions Lemma, proved in his celebrated paper of 1984. As an application, we get existence of a nontrivial bounded variation solution of a quasilinear elliptic problem involving the 1−Laplacian operator in \({\mathbb{R}^N}\), which has the lowest energy among all the radial ones. This seems to be one of the very first works dealing with stationary problems involving this operator in the whole space.

Mathematics Subject Classification (2010)

Primary 35J62 Secondary 35J93 


Bounded variation functions 1-Laplacian operator compactness with symmetry 


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Giovany M. Figueiredo
    • 1
  • Marcos T. O. Pimenta
    • 2
  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil
  2. 2.Departamento de Matemática e Computação, Faculdade de Ciências e TecnologiaUNESP - Universidade Estadual PaulistaPresidente PrudenteBrazil

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