Lie point symmetries and exact solutions of quasilinear differential equations with critical exponents

Abstract

We consider a general class of quasilinear ordinary differential equations which contains, in particular, the Lane–Emden equation, the Liouville equation, the Poisson–Boltzmann equation, equations involving the radial forms of the Laplace, p-Laplace and the k-Hessian operators. The Lie point symmetry group of these equations is calculated. Then the corresponding Noether symmetries are found and used to obtain first integrals and exact solutions of the equations with critical exponents.