Group classification of semilinear Kohn–Laplace equations

Abstract

We study the Lie point symmetries of semilinear Kohn-Laplace equations on the Heisenberg group H 1 and obtain a complete group classification of these equations.

Lie Symmetries and Criticality of Semilinear Differential Systems

“…General nonexistence results for solutions of semilinear differential inequalities on the Heisenberg group were obtained by Pokhozhaev and Veron in [33]. In [8] a complete group classification of Kohn-Laplace equations on H 1 is carried out. We observe that the Kohn-Laplace equation is formally the Euler-Lagrange equation of the functional…”

Note on Lie Point Symmetries of Burgers Equations

“…Here the summation over the repeated indexes is understood. Applications of this method in differential equations can be found in [2,3,4,5,9,10,11,12,13,16,17]. We shall not present more preliminaries concerning the Lie point symmetries of differential equations supposing that the reader is familiar with the basic notions and methods of group analysis [2,9,12].…”

Conservations laws for critical Kohn-Laplace equations on the Heisenberg group

“…In a previous work [13], we obtained the complete group classification of the following semilinear equation on the three-dimensional Heisenberg group H := H 1 :…”

“…(see [13]), all conservation laws with non-zero scaling weight can be obtained from an explicit formula of Anco [1] which uses the full admitted symmetry group of the equation. However, in our specific case, this will result in a calculation lengthier than here presented.…”

“…Actually, this is the price of generality of the modern method when applied to the considered particular equation. On the other hand, we have already granted from [13,14] the potentials ϕ and this is the reason to apply the classical approach to conservation laws.…”