of nontrivial solutions of the Robin problem − (P)

In this study, we investigate the question of nonexistence of nontrivial solutions of the Robin problem (P) � � � � � @ 2 u @x2 n P s=1 @ @ys a s(y, @u @ys ) + f(y,u) = 0 in = R × D,


Introduction
The problem of existence and nonexistence of nontrivial solutions of problems of the form −∆u + f (u) = 0 in Ω, u = 0 on ∂Ω, has been investigated by many authors under various situations.Previous works have been reported by Berestycky, Gallouet & Kavian [1] , M. J. Esteban & P. L. Lions [2], Pucci & J. Serrin [9] and Pohozaev [10].To illustrate some of the typical known results, let us consider Dirichlet problem where Ω is a connected unbounded domain of R N such as (n(x) is the outward normal to ∂Ω at the point x) Esteban & Lions [2] established that the Dirichlet problem does not have nontrivial solutions.Berestycky, Gallouet & Kavian [1] established that the problem admits a radial solution This same solution satisfies this shows that analogous Esteban-Lions result for Neumann problems is not valid.The Pohozaev identity published in 1965 for solutions of the Dirichlet problem proved absence of nontrivial solutions for some elliptic equations when Ω is a star shaped bounded domain in R n and f a continuous function on R satisfying: where J ⊂ R is unbounded interval and ω ⊂ R n domain , Haraux & Khodja [3] established under the assumption u or ∂u ∂n = 0 on ∂ (J × ω) .Then these two problems (Dirichlet and Neumann) do have only trivial solution. When and f : D×R → R a locally Lipschitz continuous function such that f (y, 0) = 0 in D, so that u = 0 is a solution of the equation We assume that u ∈ H 2 (Ω) ∩ L ∞ (Ω), and satisfies Let us denote by: The objective of this paper is to extend the results of [3], [5] to problems (1.1) − (1.2), (1.1) − (1.3) and (1.1) − (1.4).

Integral identities
We begin this section by giving an integral identity useful in the sequel.
Proof.Let H : R → R the function defined by The hypotheses on u, a i , i = 1, ..., n and f imply that H is absolutely continuous and thus differentiable almost everywhere on R, we have (2.2) Indeed a simple use of Fubini's theorem and an integration by parts yields By summing up these formulas with respect to i and substituting them in (2.2), one obtains As u satisfies equation (1.1), the above expression reduces to (2.3) EJQTDE, 2009 No. 44, p. 5 Now observe that u + ε ∂u ∂n (x, s) = 0 on ∂Ω, is equivalent to This allows to write formula (2.3) in the following form Integrating this expression,with respect to x one obtains We conclude that the constant is null which is the desired result.Proof.To prove (2.5) it suffices to show that the second term of (2.1) vanishes if u verifies (1.2) or (1.3), i.e If one supposes that u(x, s) = 0 for (x, s) ∈ R × ∂D, it is immediate, that because a i (x, 0) = 0, ∀x ∈ D, ∀i = 1, ..., n..

Main results
The goal of this section is to establish the nonexistence of nontrivial solutions to Robin problem.EJQTDE, 2009 No. 44, p. 7 Theorem 1.Let a i , i = 1, ..., n and f satisfying respectively to be a solution of (1.1) − (1.4).Then the function Proof.To begin the proof, we see that almost everywhere in Ω = R × D, we have In fact by multiplying equation ( 1 A simple use of Fubini's theorem and an integration by parts yields, Combining this formula and (2.1) we obtain Hypotheses (3.1) and (3.2) imply that This completes the proof.
Remark 1.The convexity of the function E(x) on R implies the triviality of the solution u(x, y) of the problem (1.1) − (1.4).
Theorem 2. Let the function a i , i = 1, ..., n and f be as described as in Theorem 3.1.We assume u ∈ H 3), then the function E(x) defined above is convex on R.
Proof.By similar arguments as in the proof of Theorem 3.1, we obtain Now if u (x, s) = 0 or ∂u ∂n (x, s) = 0, for (x, s) ∈ R × ∂D this formula reduces to Our assumptions on a i , and f imply the desired result.