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Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents. (English) Zbl 1061.34030
The authors investigate the second-order differential equation $$-{d\over dx} (x^\alpha\cdot\vert y'\vert^\beta\cdot y')= x^\gamma\cdot f(y)$$ with respect to Lie point symmetries, where $\alpha$, $\beta$, $\gamma$ are real constants, $x> 0$ and $f$ is a nonnegative function. There are many relations to other well-known equations, for instance to the radial forms of Laplacian and Hessian operators as well as to the Lane-Emden and Emden-Fowler equation, the Boltzmann equation and to other problems. By the evaluation and splitting of the identity corresponding to the symmetry criterion there are characterized various cases of symmetries with respect to parameter situations for $\alpha$, $\beta$, $\gamma$. The investigations are then focused to the problems of critical exponents (see {\it P. Clément, D. G. de Figueiredo} and {\it E. Mitidieri} [Topol. Methods Nonlinear Anal. 7, 133--170 (1996; Zbl 0939.35072)], to closed form solutions and to the characterization of symmetries as Noether symmetries (where the differential equation is regarded as the Euler-Lagrange equation for a functional).

##### MSC:
 34C14 Symmetries, invariants (ODE) 34A05 Methods of solution of ODE 34A34 Nonlinear ODE and systems, general
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##### References:
 [1] American Mathematical Society, MathSciNet Search. [2] Bhutani, O. P.; Vijayakumar, K.: On certain new and exact solutions of the Emden--Fowler equation and Emden equation via invariant variational principles and group invariance. J. austral. Math. soc. Ser. B 32, 457-468 (1991) · Zbl 0735.34001 [3] Yu. Bozhkov, A.C. Gilli Martins, On the symmetry group of a differential equation and the Liouville--Gelfand problem, Rend. Inst. Mat. Univ. Trieste XXXIV (2002), 103--120. · Zbl 1156.35358 [4] Bratu, G.: Sur LES équationes intégrales non linéares. Bull. soc. Math. de France 42, 113-142 (1914) · Zbl 45.0525.03 [5] Chandrasekhar, S.: An introduction to the study of stellar structure. (1958) · Zbl 0082.20502 [6] Clément, Ph.; De Figueiredo, D. G.; Mitidieri, E.: Quasilinear elliptic equations with critical exponents. Topol. methods nonlinear anal. 7, 133-164 (1996) · Zbl 0939.35072 [7] Gazzola, F.; Malchiodi, A.: Some remarks on the equation -${\Delta} u={\lambda}$ ( 1+u)p for varying domains. Comm. partial differential equations 27, 809-845 (2002) · Zbl 1010.35042 [8] Gelfand, I. M.: Some problems in the theory of quasilinear equations. Amer. math. Soc. transl. 29, 295-381 (1963) · Zbl 0127.04901 [9] Gidas, B.; Ni, W.; Nirenberg, L.: Symmetry and related problems via maximum principle. Comm. math. Phys. 68, 209-243 (1979) · Zbl 0425.35020 [10] Goenner, H.: Symmetry transformations for the generalized Lane--Emden equation q \dot{} + ${\alpha}$ (t) q \dot{} + qm = 0. Gen. relativity gravitation 33, 833-841 (2001) · Zbl 0989.83024 [11] Goenner, H.; Havas, P.: Exact solutions of the generalized Lane--Emden equation. J. math. Phys. 41, 7029-7042 (2000) · Zbl 1009.34002 [12] Joseph, D. D.; Lundgren, T. S.: Quasilinear Dirichlet problems driven by positive sources. Arch. rational mech. Anal. 48, 241-269 (1973) · Zbl 0266.34021 [13] Kara, A. H.; Mahomed, F. M.: A note on the solutions of the Emden--Fowler equation. Int. J. Non-linear mech. 28, 379-384 (1993) · Zbl 0786.34001 [14] Leach, P. G. L.: First integrals for the modified Emden equation. J. math. Phys. 26, 2510-2514 (1985) · Zbl 0587.34004 [15] Leach, P. G. L.; Maartens, R.: Self-similar solutions of the generalized Emden--Fowler equations. Int. J. Non-linear mech. 27, 575-582 (1992) · Zbl 0760.34005 [16] Liouville, J.: Sur l’équation aux dérivés partielles $\partial2$l$og{\lambda} \partial u\partial$v${\pm} {\lambda}$ a2=0. J. math. Pures appl. 18, No. 1, 71-72 (1853) [17] Logan, J. D.: Invariant variational principals. (1977) [18] P.J. Olver, Application of Lie Groups to Differential Equations, GTM 107, Springer, New York, 1986. · Zbl 0588.22001 [19] Silva, E. A. B.; Soares, S. H. M.: Liouville--Gelfand type problems for the N-Laplacian on bounded domains of RN. Ann. scuola norm. Sup. Pisa cl. Sci. 28, No. 4, 1-30 (1999) · Zbl 0943.35032 [20] Spiegel, M. R.: Schaum’s outline of theory and problems of Fourier analysis. (1974) [21] Stephani, H.: Differential equations: their solutions using symmetries. (1989) · Zbl 0704.34001 [22] Atkinson, F. V.; Peletier, L. A.: Elliptic equations with nearly critical growth. J. diff. Eq. 70, 349-365 (1987) · Zbl 0657.35058 [23] Caristi, G.; Mitidierl, E.: Nonexistence of positive solutions of quasilinier equations. Adv. diff. Eq. 2, 319-359 (1997) · Zbl 1023.34500 [24] Jacobsen, J.; Schmitt, K.: The lionville-bralu-Gelfand problem for radial operators. J. diff. Eq. 181, 283-298 (2002) · Zbl 1015.34013 [25] Knaap, M. C.; Peletier, L. A.: Quasilinear elliptic equations with nearly critical growth. Comm. part. Diff. eq. 14, 1351-1383 (1989) · Zbl 0696.35052