NONEXISTENCE RESULTS OF SOLUTIONS TO SYSTEMS OF SEMILINEAR DIFFERENTIAL INEQUALITIES ON THE HEISENBERG GROUP ABDALLAH

We establish nonexistence results to systems of differential inequalities on the ( 2 N + 1 ) -Heisenberg group. The systems considered here are of the type ( ES m ) . These nonexistence results hold for N less than critical exponents which depend on p i and γ i , 1 ≤ i ≤ m . Our results improve the known estimates of the critical exponent.


Introduction
For the reader's convenience, we recall some background facts used here.The Heisenberg group H N , whose points will be denoted by η = (x, y,τ), is the Lie group (R 2N+1 ,•) with the group operation • defined by η • η = x + x, y + ỹ,τ + τ + 2 x, ỹ − x, y , ( where •, • is the usual inner product in R N .The Laplacian ∆ H over H N is obtained, from the vector fields X i = ∂ xi + 2y i ∂ τ and Y i = ∂ yi − 2x i ∂ τ , by Observe that the vector field T = ∂ τ does not appear in (1.2).This fact makes us presume a "loss of derivative" in the variable τ.The compensation comes from the relation X i ,Y j = −4T, j,k ∈ {1, 2,...,N}. (1. 3) The relation (1.3) proves that H N is a nilpotent Lie group of order 2. Incidently, (1.3) constitutes an abstract version of the canonical relations of commutation of Heisenberg between momentums and positions.Explicit computation gives the expression (1.4) A natural group of dilatations on H N is given by whose Jacobian determinant is λ Q , where is the homogeneous dimension of H N .The operator ∆ H is a degenerate elliptic operator.It is invariant with respect to the left translation of H N and homogeneous with respect to the dilatations δ λ .More precisely, we have (1.7) It is natural to define a distance from η to the origin by In [7], Pohozaev and Véron gave another proof of the result of Birindelli et al. [1] concerning the nonexistence of weak solutions of the differential inequality They then addressed the question of nonexistence of weak solutions of the system (ES 2 ): where a i , i ∈ {1, 2}, are measurable and bounded functions defined on H N , and p i > 1 and γ i , i = 1,2, are real numbers.They showed that this system admits no solution defined in The estimates on p i , i = 1,2, are obtained using Young's inequality and are not optimal.Using the Hölder inequality, we obtain better estimates on p i , 1 ≤ i ≤ m.The same strategy is suitable to study the systems (PS m ) and (HS m ).We also studied the following systems: where the vector (X 1 ,X 2 ,...,X m ) T is the solution of (3.1), then there is no nontrivial global weak solution (u 1 ,...,u m ) of the system (PS m ).
In [2], the first author and Obeid presented results for systems of evolution type with higher-order time derivatives.Their results are the generalized versions of our previous results (Theorems 1.1 and 1.2) on (PS m ) and (HS m ).
For interesting results on elliptic equations and systems, we refer to the recent papers of Kartsatos and Kurta [3], Kurta [4,5], and Mitidieri and Pohozaev [6].
To render the presentation very clear, we start with the case of systems of two inequalities.

Systems of two inequalities
In this section, we treat the case m = 2 and consider the system (ES 2 ).
We identify points in H N with points in R 2N+1 .We also recall that the Haar measure on In the sequel, the integral R 2N+1 will be simply denoted by ; however, the measure of integration will be specified.Definition 2.1.Let a 1 and a 2 be two bounded measurable functions on R 2N+1 .A weak solution (u,v) of the system (ES 2 ) on R 2N+1 is a pair of locally integrable functions (u,v) such that for any nonnegative test function Theorem 2.2.Assume that Then there is no nontrivial weak solution (u,v) of the system (ES 2 ).
where λ 1, R > 0, and then (2.9) It follows that there is a positive constant C > 0, independent of R, such that thanks to the Hölder inequality.Setting we have where and C 1 is a positive constant independent of R. Similarly, we have where and C 2 is a positive constant independent of R.
In order to estimate the integrals Ꮽ pi,γi (R), i ∈ {1, 2}, we introduce the scaled variables Using the fact that suppϕ R ⊂ Ω R , we conclude that where (2.21) Similarly, we have where (2.23) Now, we require that σ I ≤ 0 or σ J ≤ 0, which is equivalent to (2.24) In this case, the integrals I(R) and J(R), increasing in R, are bounded uniformly with respect to R. Using the monotone convergence theorem, we deduce that |η| ).Note that instead of (2.11) we have, more precisely, (2.25) Finally, using the dominated convergence theorem, we obtain that lim which implies that v ≡ 0 and u ≡ 0 via (2.12).This contradicts the fact that (u,v) is a nontrivial weak solution of (ES 2 ), which achieves the proof.
Remark 2.3.The critical exponent Q * e can be written as (2.28) A. El Hamidi and M. Kirane 161 where the vector (X 1 ,X 2 ) T is the solution of the linear system In their paper, Pohozaev and Véron [7] showed that if then the system (ES 2 ) has no nontrivial weak solution.The condition (2.30) is equivalent to (2.31) Theorem 2.2 gives a better estimate of the exponent.Indeed, which implies that

Systems of m semilinear inequalities
In this section, we give generalizations of the last results to systems with m inequalities, m ∈ N * .Let (X 1 ,X 2 ,...,X m ) be the solution of the linear system where p i > 1 and γ i are given real numbers, i ∈ {1, 2,...,m}.
Consider the system (ES m ): where p m+1 = p 1 , γ m+1 = γ 1 .Proof.In order to simplify the proof, we treat only the case m = 3; the general case can be established in the same manner.
Let (u 1 ,u 2 ,u 3 ) be a nontrivial weak solution of (ES m ).The inequalities (3.4), with ϕ = ϕ R defined by (2.4), imply that Let then there is a positive constant C such that A. El Hamidi and M. Kirane 163 Hence, the estimates In order to estimate the expressions I i , 1 ≤ i ≤ 3, we use the scaled variables (2.18) and obtain where (3.10) Now, we require that, at least, one of σ i , 1 ≤ i ≤ 3, is less than zero, which is equivalent to Q ≤ 2 + max{X 1 ,X 2 ,X 3 }, where the vector (X 1 ,X 2 ,X 3 ) T is the solution of Following the arguments used in the proof of Theorem 2.2, we conclude that (u 1 ,u 2 ,u 3 ) ≡ (0,0,0).This ends the proof by contradiction.