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

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Abstract

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 

Keywords

Bounded variation functions 1-Laplacian operator compactness with symmetry 

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References

  1. 1.
    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. Partial Differential Equations 56(5), 143 (2017)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Anzellotti G.: The Euler equation for functionals with linear growth. Trans. Amer. Math. Soc. 290(2), 483–501 (1985)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Attouch H., Buttazzo G., Michaille G.: Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization. MPS-SIAM, Philadelphia (2006)CrossRefMATHGoogle Scholar
  4. 4.
    Balogh Z., Kristály A.: Lions-type compactness and Rubik actions on the Heisenberg group. Calc. Var. Partial Differential Equations 48(1–2), 89–109 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bartle R.: The elements of integration and Lebesgue measure. John Wiley & Sons, New York (1995)CrossRefMATHGoogle Scholar
  6. 6.
    Chang K.: Variational methods for non-differentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Clarke F.: Generalized gradients and applications. Trans. Amer. Math. Soc. 205, 247–262 (1975)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dai G.: Non-smooth version of Fountain theorem and its application to a Dirichlet-type differential inclusion problem. Nonlinear Anal. 72, 1454–1461 (2010)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Degiovanni M., Magrone P.: Linking solutions for quasilinear equations at critical growth involving the 1-Laplace operator. Calc. Var. Partial Differential Equations 36, 591–609 (2009)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Demengel F.: Functions locally almost 1-harmonic. Appl. Anal. 83(9), 865–896 (2004)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Demengel F.: On some nonlinear partial differential equations involving the 1-Laplacian and critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 4, 667–686 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Evans L., Gariepy R.: Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992)MATHGoogle Scholar
  13. 13.
    G.M. Figueiredo and M.T.O. Pimenta, Nehari method for locally Lipschitz functionals with examples in problems in the space of bounded variation functions, to appear.Google Scholar
  14. 14.
    Figueiredo G.M., Pimenta M.T.O.: Existence of bounded variation solution for a 1-Laplacian problem with vanishing potentials. J. Math. Anal. Appl. 459, 861–878 (2018)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Giusti E.: Minimal surfaces and functions of bounded variation. Birkhäuser, Boston (1984)CrossRefMATHGoogle Scholar
  16. 16.
    B. Kawohl and F. Schuricht , Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem, Commun. Contemp. Math. 9, no. 4, 525–543.Google Scholar
  17. 17.
    Kobayashi J., Ôtani M.: The principle of symmetric criticality for nondifferentiable mappings. J. Funct. Anal. 214, 428–449 (2004)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Lions P.L.: The concentration-compactness principle in the Calculus ov Variations. The Locally compact case, part 2. Annales Inst. H. Poincaré Section C 1, 223–283 (1984)CrossRefMATHGoogle Scholar
  19. 19.
    Leon S., Webler C.: Global existence and uniqueness for the inhomogeneous 1-Laplace evolution equation. NoDEA Nonlinear Differential Equations Appl. 22, 1213–1246 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Rabinowitz P.: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 43, 270–291 (1992)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Squassina M.: On Palais’ principle for non-smooth functionals. Nonlinear Anal. 74, 3786–3804 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Strauss W.A.: Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55, 149–162 (1977)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Szulkin A.: Minimax principle for lower semicontinuous functions and applications to nonlinear boundary value problems. Ann. Inst. H. Poincaré 3(2), 77–109 (1986)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© 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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