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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

$-\frac{d}{dx}\left({x}^{\alpha }·|{y}^{\text{'}}{|}^{\beta }·{y}^{\text{'}}\right)={x}^{\gamma }·f\left(y\right)$

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