zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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:
34C14Symmetries, invariants (ODE)
34A05Methods of solution of ODE
34A34Nonlinear ODE and systems, general
WorldCat.org
Full Text: DOI
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