A NONEXISTENCE RESULT FOR A NONLINEAR PDE WITH ROBIN CONDITION

Under the assumption λ > 0 and f verifying f (x, y,0)= 0 in D, 2F(x, y,u)−u f (x, y,u)≥ 0, u = 0, and if Ω= R×D, we show the convexity of function E(t)= ∫∫D |u(t,x, y)|2dxdy, where u :Ω→R is a solution of problem λ(∂2u/∂t2)−(∂/∂x)(p(x, y)(∂u/∂x))−(∂/∂y)(q(x, y)(∂u/∂y)) + f (x, y,u)= 0 in Ω, u+ ε(∂u/∂n)= 0 on ∂Ω, considered in H2(Ω)∩L∞(Ω), p,q : D→R are two nonnull functions on D, ε is a positive real number, and D = ]a1,b1[ ×]a2,b2[, (F(x, y,s)= ∫ s 0 f (x, y, t)dt).


Introduction
In this paper we consider the question of absence of nontrivial solutions of the following boundary value problem: are two continuous functions > 0 or < 0 in D, ε is a positive real number, and f is a locally Lipschitz continuous function such that f (x, y,0) = 0 in D, (1.3) so that is a solution of problem (P).This problem is considered in the Sobolev space with This question has interested much researchers and a significant number of works, were carried out.We quote by the way of examples the works of De Figueiredo and Jianfu [2], Esteban and Lions [3], Pucci and Serrin [6], Pohožaev [5], and Van der Vorst [7].
Esteban and Lions show that the Dirichlet problem where Ω is a connected unbounded domain of R N such that does not have nontrivial solution.Berestycky et al. [1] established that the problem Brahim Khodja 3 admits a radial solution.This solution satisfies ( 1.11)This shows that the analog of the Dirichlet problem for the Neumann problem is not true.The problem (1.12) considered in the Sobolev space is still open in These considerations motivated us to explore more this question of the absence of nontrivial solutions which presents a double interest.It makes it possible to solve an open problem, and to ensure then that the single solution is the null solution.
The aim of this work is to extend the results of [4] to problem (P).We prove in Section 2 a Pohožaev-type identity.
In Section 3, we combine Theorem 2.1 with other results to obtain for the semilinear elliptic problems a corollary of nonexistence of solutions.For the semilinear hyperbolic problems, we obtain in the same section an interesting result which shows that the Klein-Gordon align does not have nontrivial solutions.
Finally in Section 4, we give some examples to illustrate Theorems 3.1 and 3.5.
Let us denote by where where (1.19)

Integral identity
We now give an integral identity in the form of theorem.
Theorem 2.1.Let u be an element of H 2 (Ω) ∩ L ∞ (Ω), a solution of problem (P), then for each t ∈ R and ε > 0, Proof.For t ∈ R, we consider a function K defined by 2) The hypotheses on u, p, q, and f imply that K is absolutely continuous and thus differentiable almost everywhere on R; we have ( Replacing in (2.3), we find Let us write on ∂Ω the expression u + ε∂u/∂n = 0 in an equivalent way: ( (2.9) K , derivative of K, verifies therefore (2.12) The constant is null; this shows the theorem.

Main results
The parameter λ plays in fact an important part as it allows problem (P) to be dealt with in two manners according to whether its value is positive or negative.
Proof.Applying formula (2.1) we immediately obtain The following theorem gives a nonexistence result if f satisfies another type of nonlinearity.
be a solution of problem (P), p(x, y), q(x, y) > 0 or < 0 in D, (3.4) and f a function verifying Then the function

Semilinear elliptic problems.
For the elliptic case, there is a nonexistence result which is stated as follows.
Theorem 3.5.Let be a solution of (P), λ < 0, and f satisfying Then the function E(t) defined in Theorem 3.2 is convex on R.
Proof.The proof is similar to that of Theorem 3.2.

Examples
In this paragraph we will discuss some examples to demonstrate the use of Theorems 3.1 and 3.2.Then problem does not have nontrivial solutions.In the following situations: we can see that in the first case and in the second case  does not have nontrivial solutions in H 2 (Ω) ∩ L ∞ (Ω).