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Group Classification of Semilinear Kohn-Laplace Equations

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.


Introduction
The Heisenberg group H n topologically is the real vector space R 2n+1 . Its Lie group structure is determined by the product (x, y, t)(x 0 , y 0 , t 0 ) = (x + x 0 , y + y 0 , t + t 0 + 2 where (x, y, t), (x 0 , y 0 , t 0 ) ∈ R n × R n × R = H n . It is easy to verify that the operators where i = 1, 2, ..., n, form a basis of the left-invariant vector fields on H n and satisfy the following commutation relations: These formulae present in an abstract form the commutation relations for the quantummechanical position and momentum operators in n−dimensional configuration space. This justifies the name Heisenberg group.
In the last few decades the Heisenberg group H n was intensively and extensively studied by a considerable number of authors using methods and approaches which come from algebraick and differential geometry, real and complex analysis, mathematical physics and applications. A big part of the corresponding works treats partial differential equations on H n . In this regard various authors have obtained existence and nonexistence results for equations involving Kohn-Laplace operators. Recall that the Kohn-Laplace operator ∆ H n is the natural subeliptic Laplacian on H n defined by Although there are similarities between ∆ H n and the classical Laplacian they are essentially different. E.g. the Kohn-Laplace operator is not a strongly elliptic operator. It is a typical representative of the hypoeliptic operators ( [19]). (Since the study of hypoellipticity properties is not subject of this paper we shall not comment more on this point.) In [18] Garofalo and Lanconelli established existence, regularity and nonexistence results for the Kohn-Laplace equation in an open bounded or unbounded subset of H n with homogeneous Dirichlet boundary condition. One of the motivations to study such semilinear equations is the fact that they may arise as Euler-Lagrange equations in some variational problems on Cauchy-Riemann (CR) manifolds as in the works of Jerison and Lee [25,26] on the CR Yamabe problem. The existence of weak solutions is proved in [18] provided the nonlinear term satisfies some growth conditions of the form f (u) = o(|u| (Q+2)/(Q−2) ) as |u| → ∞, where Q = 2n + 2 is the so-called homogeneous dimension of H n ( [15]). The exponent (Q + 2)/(Q − 2) is the critical exponent for the Stein's Sobolev space ( [26]). The nonexistence results follow from remarkable Pokhozhaev Identities established in [18] for the solutions of Kohn-Laplace equations on the Heisenberg group. The Dirichlet problem for the Kohn Laplacian on H n was studied before by Jerison in [23,24]. See also [4] for existence of classical nonnegative solutions of semilinear Kohn-Laplace equations. General nonexistence results for solutions of semilinear differential inequalities on the Heisenberg group were obtained by Pokhozhaev and Veron in [29]. Since there is a huge number of works dedicated to Heisenberg groups (see [1]) and the study of PDE on H n , in order not to increase the volume of this paper, we shall not present here further details, directing the interested reader to the already cited works as well as to [2,3,5,6,14,15,20,21] and the references therein. The purpose of the present paper is to enlighten the properties of the Kohn-Laplace equations from the point of view of the S. Lie Symmetry Theory, which to our knowledge has not been previously done. We shall obtain complete group classification of semilinear partial differential equations on H 1 of the following form where ∆ H 1 is the Kohn-Laplace operator on H 1 and f is a generic function. The importance of group classification of differential equations was first emphasized by Ovsiannikov in 1950s-1960s, when he and his school began a systematic research program of successfully applying modern group analysis methods to wide range of physically important problems. Following Olver ([27], p. 182), we recall that to perform a group classification on a differential equation involving a generic function f consists of finding the Lie point symmetries of the given equation with arbitrary f , and, then, to determine all possible particular forms of f for which the symmetry group can be enlarged. It is worth observing that for problems which arise from physics, quite often there exists a physical motivation for considering such specific cases.
The Heisenberg group H 1 itself possesses the rich properties of H n (see [3]) and the calculations of the symmetry group of this model problem give insights for the general case n > 1. For this reason, and for the sake of simplicity and clarity we restrict ourselves to H 1 .
We write the Kohn-Laplace operator where and Then the equation (1) for u = u(x, y, t) in more details reads We shall not present preliminaries concerning Lie point symmetries of differential equations supposing that the reader is familiar with the basic notions and methods of contemporary group analysis [8,22,27,28].
The main result in this paper is the following Theorem The widest Lie point symmetry group of the Kohn-Laplace equation (1) with an arbitrary f (u) is determined by the operators that is, by a translation in t, a rotation in the x-y plane and the generators of right multiplication in the Heisenberg group H 1 . For some special choices of the right-hand side f (u) it can be extended in the cases listed below. We shall write only the generators additional to (6).
(ii) If f (u) = c = const, then this case is reduced to (i) by the change u = v + cx 2 /2.
(v) If f (u) = k.e u then the operator generates a sub-group of the Lie point symmetry group of (1).
This classification is similar to that for semilinear equations in R n involving Laplace or polyharmonic operators [30]. We also observe that for power nonlinearity f (u) = ku p exactly in the critical case p = 3 = (Q+2)/(Q−2), Q = 2.1+2 = 4 being the homogeneous dimension of H 1 , the symmetry group is expanded by three aditional generators (see (iv) of the main theorem). This fact suggests that in the critical case maybe there are further properties as pointed out in [9,10] regarding other differential equations. This is our motivation to use the above group classification in three subsequent papers [11,13,12]. In [11] we study the variational properties of Kohn-Laplace equations and we find out which of the already found Lie point symmetries are variational/divergence symmetries. Further in [12] we establish the corresponding conservation laws via the Noether Theorem. In [13] we discuss the invariant solutions of various Kohn-Laplace equations on the Heisenberg group.
The group classification of Kohn-Laplace equations on the Heisenberg group H n , n > 1, will be treated elswhere.
This paper is organized as follows. In the next section we obtain the determining equations for the Lie point symmetries of the equation (1). This process is essentially simplified by the use of two theorems of Bluman [7,8]. Then in section 3 we obtain some formulae which are consequences of the determining equations. They are used in the proof of the main theorem, given in sections 4-9.

The determining equations
In this section we obtain the determining equations for a Lie point symmetry of the Kohn-Laplace equation (1) with infinitesimal generator To begin with, we observe that the symmetry calculation is drastically simplified if we apply two theorems of Bluman [7,8]. Indeed, Theorem 4.2.3-1, [8], p. 174, implies that ξ, φ and τ do not depend on u. Then by Theorem 4.2.3-6, [8], p. 175, we conclude that η is a linear function of u. Therefore the infinitesimals are of the following form where α = α(x, y, t) and β = β(x, y, t) are functions to be determined.
We denote The equation (1) admits the symmetry (14) if and only if is the second order extension of S ( [8,27]). Then the symmetry condition can be written as when H = 0. (The subscripts denote partial derivatives, e.g. u x = ∂u ∂x . Only in the extension coefficients like η (j) xt the subscripts mean indices. We also suppose that the considered functions are sufficiently smooth in order that the derivatives we write to exist.) Further, using the corresponding formulae for the extended infinitesimals ( [8,27]) we calculate Now substituting (18) − (25) into the symmetry condition (17), after some tedious work, we obtain when H = 0. Then, expressing u xx from (5) and substituting in (26), we obtain an identity for all values of (x, y, t, u, u x , u y , u t , u xy , u xt , u yy , u yt , u tt ). Equating to zero the coefficients of the derivatives of u and the free term, we obtain the following nine determining equations: where the operators X and Y are defined by (3) and (34). Another straightforward calculation shows that (31) also follows from these equations. Therefore there are seven independent determining equations which in terms of the operators X and Y can be written in the following simplified form: We conclude this section by noting that the system of two equations (36) -(37) may be considered as a Heisenberg group generalization of the Cauchy-Riemann equations.
3 Some consequences of the determining equations Proposition 1. If the infinitesimals ξ and φ satisfy (36) − (37) then and Proof. We apply X to equation (37), Y to equation (36) and subtract the resulted equations. In this way we obtain and ∆ H 1 = X 2 + Y 2 . The equation (44) can be derived in an analogous way.

The Lie point symmetries for arbitrary f (u)
In this section we prove the main theorem for general right-hand side of the Kohn-Laplace equation (1).
Since f (u) is an arbitrary function, then α = β = 0 by (40). Thus the equations (38) and (39) imply that Then by (51), (43) and (44)  which implies that ξ depends only on y since α = 0 and ξ does not depend on t. From the equation (36) for ξ(y) and φ(x, y) it follows that φ depends only on x. Further, from (51), we obtain that ξ = a 1 y + a 2 , where A, a 1 , a 2 , a 3 are arbitrary constants. Now from (37) we get that A = −a 1 , that is Substituting ξ and φ into (41) and (42) gives Applying Y and X to the latter two equations, subtracting and using the commutator [X, Y ] = −4∂ t we obtain that τ does not depend on t. Therefore from which we conclude easily that τ = 2a 3 x − 2a 2 y + a 4 where a 4 is another arbitrary constant. In this way which proves the first statement of the main theorem.

The Lie point symmetries for f (u) = ke u
In this section we prove the item (v) of the main theorem. We substitute f (u) = ke u into (40): and From (53) and (46), (47) it follows that ξ = ξ(x, y) and φ = φ(x, y). From (54): and hence β = β(x, y). Further, the relation (48) implies that τ t = 2ξ x since ξ and φ do not depend on t. Therefore there exists a function h(x, y) such that We observe that he right-hand side of the equation (41) does not depend on t, while the left-hand side is 2t ξ xx + h x + 4yξ x .
6 The Lie point symmetries for f (u) = ku p In this section we prove the main theorem in the case of nonlinearity of power type f (u) = ku p . We suppose that p = 0, p = 1, p = 2, p = 3. We do not consider p = 2 since in this case by a nonexistence result of Pokhozhaev and Veron [29] there is no solution of the corresponding Kohn-Laplace equation even in a very weak sense. The case p = 3 will be treated in the next section. The case p = 1 will be studied in section 9. Finally, if p = 0 this is the item (ii), which is reduced to (i) as stated in the theorem. By (40) we have Hence β = 0, ∆ H 1 α = 0 and By (49) and (61) it follows that (p − 3)α t = 0.
From (36), (85) and (84) we have that Clearly, from (86) and (87), the function ϕ is of the following form: where k 1 , k 2 , k 3 , k 4 are arbitrary constants. In this way Substituting ξ from (89) and ϕ from (88) into (80), and integrating with respect to y, we obtain where h 1 is a function of x only. Analogously, from (81) we find where h 2 is a function of y only. After a substitution of ξ and φ from (89) and (90) with B 1 and B 2 given by (91) and (92), into (36), we obtain Obviously, the last two equations can be easily integrated. In this way we find the functions h 1 , h 2 , and hence the functions B 1 and B 2 . Summarizing, we have found ξ = − 1 6 (k 1 x + k 2 ) t + 1 6 k 1 x 2 y + 1 6 k 1 y 3 + 2 3 k 2 xy − 1 6 k 3 x 2 + k 2 2 y 2 +k 4 y + k 5 x + k 6 , φ = − 1 It remains to find τ . In order to do this, we substitute (93) into (48) and obtain and the problem is reduced to the problem of finding the function N in (94). Substituting (93) and (94) into equations (41) and (42), after some work, we finally obtain The latter system can be easily solved. After renaming the constants, we have completing the proof of item (iv) of the main theorem.
We observe that the dilation Z is included in (95). Indeed, it corresponds to the constant a 4 .

The Lie point symmetries for f (u) = 0
The proof of item (i) of the main theorem is presented in this section. In order not to increase the volume of this paper, some of the calculations will be sketched, leaving the details to the interested reader.
This completes the proof of the theorem.