Absence of Nontrivial Solutions for a Class of Partial Differential Equations and Systems in Unbounded Domains

In this paper, we are interested on the study of the nonexistence of non-trivial solutions for a class of partial differential equations, in unbounded domains. This leads us to extend these results to m-equations systems. The method used is based on energy type identities.


Introduction
The study of the nonexistence of nontrivial solutions of partial differential equations and systems is the subject of several works of many authors, by using various methods to obtain the necessary and sufficient conditions, so the studied problems admit only the null solutions.The works of Esteban & Lions [2], Pohozaev [6] and Van Der Vorst [7], contains results concerning the semilinear elliptic equations and systems.A semilar result can be found in [4], where studied equations one of the form In this work similar results for a class of the partial differential equations and systems were also obtained.Let us consider the following problem in H 2 (R×Ω)∩L ∞ (R×Ω), Ω a bounded domain of R n , for a function λ : R → R, not changing sign and p : R × Ω → R also had not changing sign.
We use the notations , the norm of u in H, Let L be the operator defined by and satisfies the equation under the boundary conditions We extend the above result of (1.1) to the system of m-equations of the form Let L k be the operators defined by we assume that are solutions of the system ) According to the sign of λ, this type of problems comprises equations of both hyperbolic or elliptic type.Our proof is based on energy type identities established in section 2, which make it possible to obtain the main nonexistence result in section 3.In section 4 we apply the results to some examples.

Identities of energy type
In this section, we give essential lemmas for showing the main result of this paper.
Lemma 1 Let λ and p satisfy Then the following energ identity, holds for any solution of the Robin problem of (1.2) In addition, consider the functions where Φ and Ψ are of class C 1 ,and Define the function K : R → R by The function K is absolutely continuous and differentiable in R, and Because u is solution of (1.2) − (1.3), we deduce that Also The remainder of the proof is similar to that of Lemma 1.
Lemma 3 Let λ and p k satisfy (2.4) Then any solutions of the system (1.7)−(1.8)satisfies for all t ∈ R, the following energetic identity Lemma 4 Let λ and p k verify (2.5).Then the solutions of the systems (1.7) − (1.9) or (1.7) − (1.10) , satisfies for all t ∈ R, the following estimate (2.6) Proof.Let us define the function K m : R → R by where the functions Ψ m and Φ m are defined as follows the rest of the proof is similar to the proofs of the preceding lemmas.

The main Result
Theorem 1 Let us suppose that λ, F and f verify Multiplying equation (1.1) by u and integrating the new equation on Ω, we obtains Using identity (2.2), we have EJQTDE, 2009 No. 27, p. 6 If λ (t) > 0, the assumption (3.1) implies that We conclude that If λ (t) < 0, we deduce by the same manner that u = 0 in R × .Proof.Identical to that of Theorem 1.
and (2.5) holds.Then the system (1.7) − (1.8) admit only the null solutions.Proof.Multiplying equation (1.6) by u k and integrating the new equation on Ω, one obtains EJQTDE, 2009 No. 27, p. 7 The sum on k from 1 to m gives By using identity (2.6), we deduce that Then the assumption (3.4) gives the result.
Theorem 4 Let λ, p and F verify Proof.Assumptions (3.4) and equality (2.3) allow is But the following condition is necessary Proof.Similar to that of Theorem 2.
Remark 1 Note that one can apply these results in the field R + × Ω, with the condition u (0, x) = 0, ∀x ∈ Ω.
Then the problem defined by admits anly the null solution.
In this case it suffice to check that a 2 , b 2 [ and this equation does not admit nontrivial solutions if the following conditions holds f (0) = 0, 2F (u) − uf (u) ≤ 0.

. 1 )
holds.Then the problem (1.2) − (1.3) admit only the null solution.Proof.Let us define the function E by
≤ m, where f k : Ω × R m → R,are real continuous functions, locally Lipschitz in u i , verifing