*(English)*Zbl 1061.34030

The authors investigate the second-order differential equation

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 *P. Clément, D. G. de Figueiredo* and *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 |