Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 1 (2005), 022, 12 pages      nlin.SI/0511058      https://doi.org/10.3842/SIGMA.2005.022

Noether Symmetries and Critical Exponents

Yuri Bozhkov
Departamento de Matemática Aplicada - DMA, Instituto de Matemática, Estatistica e Computação Cientí fica - IMECC,
Universidade Estadual de Campinas - UNICAMP, C.P. 6065, 13083-970 - Campinas - SP, Brasil

Received October 03, 2005, in final form November 19, 2005; Published online November 25, 2005

Abstract
We show that all Lie point symmetries of various classes of nonlinear differential equations involving critical nonlinearities are variational/divergence symmetries.

Key words: divergence symmetry; critical exponents.

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References

  1. Bluman G.W., Kumei S., Symmetries and differential equations, New York, Springer, 1989.
  2. Bratu G., Sur les équationes intégrales non linéares, Bull. Soc. Math. de France, 1914, V.42, 113-142.
  3. Bozhkov Y.D., Divergence symmetries of semilinear polyharmonic equations involving critical nonlinearities, submitted.
  4. Bozhkov Y.D., Gilli Martins A.C., On the symmetry group of a differential equation and the Liouville-Gelfand problem, Rend. Istit. Mat. Univ. Trieste, 2002, V.34, 103-120.
  5. Bozhkov Y.D., Gilli Martins A.C., Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents, Nonlinear Anal., 2004, V.57, N 5-6, 773-793.
  6. Bozhkov Y.D., Gilli Martins A.C., Lie point symmetries of the Lane-Emden system, J. Math. Anal. Appl., V.294, 2004, 334-344.
  7. Clément P., de Figueiredo D.G. , Mitidieri E., Quasilinear elliptic equations with critical exponents, Topol. Methods Nonlinear Anal., 1996, V.7, 133-164.
  8. Gelfand I.M., Some problems in the theory of quasilinear equations, Amer. Math. Soc. Transl., 1963, V.29, 295-381.
  9. Gidas B., Spruck J., Global and local behaviour of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 1981, V.34, 525-598.
  10. Gidas B., Ni W., Nirenberg L., Symmetry and related problems via maximum principle, Comm. Math. Phys., 1979, V.68, 209-243.
  11. Hulshof J., van der Vorst R.C.A.M., Assymptotic behavior of ground states, Proc. Amer. Math. Soc., 1996, V.124, 2423-2431.
  12. Lions P.L., The concentration-compactness principle in the calculus of variations, part 1, Rev. Mat. Iberoam., 1985, V.1, 145-201.
  13. Mitidieri E., A Relich type identity and applications, Commun. Partial Differential Equations, 1993, V.18, 125-151.
  14. Mitidieri E., Private communication, January 1995.
  15. Mitidieri E., Nonexistence of positive solutions of semilinear elliptic systems in RN, Differential Integral Equations, 1996, V.9, 465-479.
  16. Mitidieri E., Private communication, January 2002.
  17. Olver P.J., Applications of Lie groups to differential equations, New York, Springer, 1986.
  18. Serrin J., Zou H., Non-existence of positive solutions of semilinear elliptic systems, Discourses in Mathematics and its Applications, Vol. 3, Department of Mathematics, Texas A&M University, College Station, Texas, 1994, 55-68.
  19. Serrin J., Zou H., Non-existence of positive solutions of the Lane-Emden systems, Differential Integral Equations, 1996, V.9, 635-653.
  20. Serrin J., Zou H., Existence of positive solutions of the Lane-Emden systems, Atti Sem. Mat. Fis. Univ. Modena, 1998, V.46, suppl., 369-380.
  21. Svirshchevskii S.R., Symmetry of nonlinear elliptic equations with critical values of parameters, Preprint 118, Moscow, Keldysh Institute of Applied Mathematics, Academy of Sciences USSR, 1989.
  22. Svirshchevskii S.R., Symmetries of nonlinear polyharmonic equations, Preprint 11, Moscow, Institute of Mathematical Modelling, Academy of Sciences USSR, 1993.
  23. Svirshchevskii S.R., Group classification of nonlinear polyharmonic equations and their invariant solutions, Diff. Equations, 1993, V.29, 1538-1547 (in Russian: Diferentsial'nye Uravneniya, 1993, V.29, N 10, 1772-1781).

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