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Quasilinear elliptic systems involving the 1-Laplacian operator with subcritical and critical nonlinearities

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Abstract

In this paper, we study some systems of elliptic PDEs involving the \(1-\)Laplacian operator. In the first one, we deal with the subcritical regime, while in the second, we study a system with nonlinearities with critical growth. The approach is based on an approximation argument, in which the solutions are obtained as the limit of related problems with the \(p-\)Laplacian operator. In order to overcome the lack of compactness in the critical case, a version of the Concentration of Compactness Principle of Lions is proved.

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References

  1. Alves, C.O., de Morais Filho, D.C., Souto, M.A.S.: On systems of elliptic equations involving subcritical or critical Sobolev exponents Nonlinear Analysis, Elsevier Science Publishing Company , 42, 771–787 (2000)

  2. Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford (2000)

    Book  Google Scholar 

  3. Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: The Dirichlet problem for the total variation flow. J. Functional Anal. 180(2), 347–403 (2001)

    Article  MathSciNet  Google Scholar 

  4. Andreu, F., Ballester, C., Caselles, V., Mazón, J. M.: Minimizing total variation flow, C. R. Acad. Sci., Paria, Sr. I, Math., Vol. 331, No. 11, 867 - 872 (2000)

  5. Andreu, F., Ballester, C., Caselles, V., Mazón, J.M.: Minimizing total variation flow. Differential Integral Equ. 14(3), 321–360 (2001)

    MathSciNet  Google Scholar 

  6. Andreu, F., Caselles, V., Mazón, J.M.: Parabolic quasilinear equations minimizing linear growth functionals, Progress in mathematics, 233. Birkhäuser Verlag, Basel (2004)

    Book  Google Scholar 

  7. Anzellotti, G.: Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 4(135), 293–318 (1983)

    Article  MathSciNet  Google Scholar 

  8. Boccardo, L., Figueiredo, de, DG: Some remarks on a system of quasilinear elliptic equations . Birkhäuser, 309–323 (2002)

  9. Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Commun. Pure Appl Math. 36, 437–477 (1987)

    Article  MathSciNet  Google Scholar 

  10. Chen, G., Frid, H.: Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147(2), 89–118 (1999)

    Article  MathSciNet  Google Scholar 

  11. Degiovanni, M.: Paola, Magrone: linking solutions for quasilinear equations at critical growth involving the “1-Laplace” operator. Calculus Variations Partial Differential Equ. 36, 591–609 (2009)

  12. Françoise, D.: On some nonlinear partial differential equations involving the 1-Laplacian and critical Sobolev exponent ESAIM: control. Optim. Calculus of Variations, EDP Sci. 4, 667–686 (1999)

    Article  Google Scholar 

  13. Françoise, D.: Functions locally almost 1-harmonic. Appl. Anal.Taylor & Francis 83, 865–896 (2004)

    MathSciNet  Google Scholar 

  14. Azorero, G.J.P., Peral Alonso, I.: Existence and nonuniqueness for the p-Laplacian communications in partial differential equations. Taylor & Francis 12, 126–202 (1987)

    Google Scholar 

  15. Kawohl, B.: On a family of torsional creep problems. J. Reine Angew. Math. 410, 1–22 (1990)

    MathSciNet  Google Scholar 

  16. Pierre-Louis, L.: The concentration-compactness principle in the calculus of variations The limit case part 1. Revista matemática iberoamericana 1, 145–201 (1985)

    Article  MathSciNet  Google Scholar 

  17. Salas, M.: Alexis and Segura de León, Sergio Elliptic equations involving the 1-Laplacian and a subcritical source term Nonlinear Analysis. Elsevier 168, 50–66 (2017)

    Google Scholar 

  18. Filho, M.: DC de and Souto Systems ofp-laplacean equations involving homogeneous nonlinearities with critical sobolev exponent degrees Communications in partial differential equations. Taylor & Francis 24, 1537–1553 (1999)

    Google Scholar 

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Acknowledgements

Marcos T.O. Pimenta is partially supported by FAPESP 2021/04158-4, CNPq 304765/2021-0 and FAPDF, Brazil. Yino Beto Cueva Carranza is supported by CAPES/Brazil 001.

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  1. Departamento de Matemática, Intituto de Biociências, Letras e Ciências Exatas, Universidade Estadual Paulista - Unesp, São José do Rio Preto, SP, CEP: 15054-000, Brazil

    Yino B. Cueva Carranza

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

    Marcos T. O. Pimenta

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  1. Yino B. Cueva Carranza
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  2. Marcos T. O. Pimenta
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Correspondence to Marcos T. O. Pimenta.

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Carranza, Y.B.C., Pimenta, M.T.O. Quasilinear elliptic systems involving the 1-Laplacian operator with subcritical and critical nonlinearities. Rend. Circ. Mat. Palermo, II. Ser 73, 1037–1058 (2024). https://doi.org/10.1007/s12215-023-00969-2

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