Geodesic
Lie Symmetries and Criticality of Semilinear Differential Systems
Symmetry, integrability and geometry: methods and applications, Tome 3 (2007)

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We discuss the notion of criticality of semilinear differential equations and systems, its relations to scaling transformations and the Noether approach to Pokhozhaev's identities. For this purpose we propose a definition for criticality based on the S. Lie symmetry theory. We show that this definition is compatible with the well-known notion of critical exponent by considering various examples. We also review some related recent papers.
Keywords: Pokhozhaev identities; Noether identity; critical exponents. 
Yuri Bozhkov; Enzo Mitidieri. Lie Symmetries and Criticality of Semilinear Differential Systems. Symmetry, integrability and geometry: methods and applications, Tome 3 (2007). http://geodesic.mathdoc.fr/item/SIGMA_2007_3_a52/
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[1] Anco S. C., Ivanova N. M., “Conservation laws and symmetries of semilinear radial wave equations”, J. Math. Anal. Appl., 332:2 (2007), 863–876 ; math-ph/0608037 | DOI | MR | Zbl

[2] Anco S. C., Liu S., “Exact solutions of semilinear radial wave equations in $n$ dimensions”, J. Math. Anal. Appl., 297 (2004), 317–342 ; math-ph/0309049 | DOI | MR | Zbl

[3] Baouendi M. S., “Sur une classe d'operateurs elliptique dégénérés”, Bull. Soc. Math. France, 95 (1967), 45–87 | MR | Zbl

[4] Bliss G., “An integral inequality”, J. London Math. Soc., 5 (1930), 40–46 | DOI | Zbl

[5] Bluman G. W., Kumei S., Symmetries and differential equations, Springer, New York, 1989 | MR | Zbl

[6] Bozhkov Y. D., “Noether symmetries and critical exponents”, SIGMA, 1 (2005), 022, 12 pp., ages ; nlin.SI/0511058 | MR | Zbl

[7] Bozhkov Y. D., “Divergence symmetries of semilinear polyharmonic equations involving critical nonlinearities”, J. Differential Equations, 225 (2006), 666–684 | DOI | MR | Zbl

[8] Bozhkov Y. D., Freire I. L., “Group classification of semilinear Kohn–Laplace equations”, Nonlinear Anal., 68:9 (2008), 2552–2568 | DOI | MR | Zbl

[9] Bozhkov Y. D., Freire I. L., Divergence symmetries of critical Kohn–Laplace equations on Heisenberg groups, Diff. Uravn., to appear

[10] Bozhkov Y. D., Gilli Martins A. C., “On the symmetry group of a differential equation and the Liouville–Gelfand problem”, Rendiconti dell'Istituto di Matematica dell'Università di Trieste, 34 (2002), 103–120 | MR | Zbl

[11] Bozhkov Y. D., Gilli Martins A. C., “Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents”, Nonlinear Anal., 57 (2004), 773–793 | DOI | MR

[12] Bozhkov Y. D., Gilli Martins A. C., “Lie point symmetries of the Lane–Emden system”, J. Math. Anal. Appl., 294 (2004), 334–344 | DOI | MR | Zbl

[13] Bozhkov Y. D., Mitidieri E., “The Noether approach to Pokhozhaev's identities”, Mediterr. J. Math., 4:4 (2007), 383–405 | DOI | MR | Zbl

[14] Caffarelli L. A., Gidas B., Spruck J., “Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth”, Comm. Pure Appl. Math., 42 (1989), 271–297 | DOI | MR | Zbl

[15] Clément P., de Figueiredo D. G., Mitidieri E., “Quasilinear elliptic equations with critical exponents”, Topol. Methods Nonlinear Anal., 7 (1996), 133–170 | MR | Zbl

[16] Clément P., Felmer P., Mitidieri E., “Homoclinic orbits for a class of infinite-dimensional Hamiltonian systems”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 367–393 | MR | Zbl

[17] Clément P., van der Vorst R. C. A. M., “On the non-existence of homoclinic orbits for a class of infinite-dimensional Hamiltonian systems”, Proc. Amer. Math. Soc., 125 (1997), 1167–1176 | DOI | MR | Zbl

[18] D'Ambrosio L., “Hardy inequalities related to Grushin type operators”, Proc. Amer. Math. Soc., 132 (2004), 725–734 | DOI | MR

[19] Garofalo N., Lanconelli E., “Existence and nonexistence results for semilinear equations on the Heisenberg group”, Indiana Univ. Math. J., 41 (1992), 71–98 | DOI | MR | Zbl

[20] Gidas B., Spruck J., “Global and local behavior of positive solutions of nonlinear elliptic equations”, Comm. Pure Appl. Math., 34 (1981), 525–598 | DOI | MR | Zbl

[21] Grushin V. V., “On a class of hypoelliptic operators”, Math. Sbornik USSR, 12:3 (1970), 458–476 | DOI

[22] Guedda M., Veron L., “Quasilinear elliptic equations involving critical Sobolev exponents”, Nonlinear Anal., 13 (1989), 879–902 | DOI | MR | Zbl

[23] Ibragimov N. H., “Noether's identity”, Dinamika Sploshnoy Sredy, 38 (1979), 26–32 (in Russian) | MR

[24] Ibragimov N. H., Transformation groups applied to mathematical physics, D. Reidel Publishing Co., Dordrecht, 1985 | MR | Zbl

[25] Mitidieri E., A Rellich identity and applications, 25, Univ. Udine, 1990, 1–35

[26] Mitidieri E., “A Rellich type identity and applications”, Commun. Partial Differential Equations, 18 (1993), 125–151 | DOI | MR | Zbl

[27] Mitidieri E., “Nonexistence of positive solutions of semilinear elliptic systems in $R^N$”, Differential Integral Equations, 9 (1996), 465–479 | MR | Zbl

[28] Mitidieri E., Pokhozhaev S. I., “A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities”, Proc. Steklov Inst. Math., 234, 2001, 1–383 | MR

[29] Olver P. J., Applications of Lie groups to differential equations, Springer, New York, 1986 | MR

[30] Ovsiannikov L., Group analysis of differential equations, Academic Press, London, 1982 | MR | Zbl

[31] Soviet Math. Dokl., 6 (1965), 1408–1411 | MR | Zbl

[32] Math. USSR Sbornik, 11:2 (1970), 171–188 | DOI | MR | MR | Zbl

[33] Pohozaev S. I., Veron L., “Nonexistence results of solutions of semilinear differential inequalities on the Heisenberg group”, Manuscripta Math., 102 (2000), 85–99 | DOI | MR | Zbl

[34] Pucci P., Serrin S., “A general variational identity”, Indiana Univ. Math. J., 35 (1986), 681–703 | DOI | MR | Zbl

[35] Pucci P., Serrin S., “Critical exponents and critical dimensions for polyharmonic operators”, J. Math. Pures Appl. (9), 69 (1990), 55–83 | MR | Zbl

[36] Serrin J., Zou H., “Non-existence of positive solutions of semilinear elliptic systems”, Discourses Math. Appl., 3 (1994), 55–68 | MR | Zbl

[37] Serrin J., Zou H., “Non-existence of positive solutions of the Lane–Emden systems”, Differential Integral Equations, 9 (1996), 635–653 | MR | Zbl

[38] Serrin J., Zou H., “Existence of positive solutions of the Lane–Emden systems”, Atti Sem. Mat. Fis. Univ. Modena, 46, suppl. (1998), 369–380 | MR | Zbl

[39] van der Vorst R. C. A. M., “Variational identities and applications to differential systems”, Arch. Rational Mech. Anal., 116 (1992), 375–398 | DOI | MR