Electronic Journal of Differential Equations,
Vol. 2009(2009), No. 02, pp. 1-11.

Title: Nonexistence results for semilinear systems in unbounded domains
   
Authors: Brahim Khodja  (Badji Mokhtar Univ., Annaba, Algeria)
         Abdelkrim Moussaoui (Badji Mokhtar Univ., Annaba, Algeria)

Abstract:
 This paper concerns the non-existence of nontrivial solutions for the
 semi-linear system of gradient type
$$\displaylines{
\lambda \frac{\partial ^{2}u_{k}}{\partial t^{2}}
-\sum_{i=1}^n \frac{\partial }{\partial x_{i}}(p_{i}(x)\frac{
\partial u_{k}}{\partial x_{i}})+f_{k}(x,u_{1},\dots ,u_{m})
=0\quad \text{in }\Omega ,\; k=1,\dots ,m
}$$
 with Dirichlet, Neumann or Robin boundary conditions. The functions 
 $f_{k}:\mathcal{D}\times \mathbb{R}^{m}\to \mathbb{R}$
 $(k=1,\dots ,m)$ are locally Lipschitz continuous and satisfy
$$
2H(x,u_{1},\dots ,u_{m})-\sum_{k=1}^m
u_{k}f_{k}(x,u_{1},\dots ,u_{m})\geq 0\quad (\text{resp.}\leq 0)
$$
for $\lambda >0$ (resp. $\lambda <0$). We establish the non-existence of
nontrivial solutions using Pohozaev-type identities.
Here $u_{1},\dots ,u_{m}$ are in $H^{2}(\Omega )\cap L^{\infty }(\Omega )$,
$\Omega =\mathbb{R}\times \mathcal{D}$ with
$\mathcal{D}=\prod_{i=1}^n  (\alpha _{i},\beta _{i})$ and
$H\in \mathcal{C}^{1}(\overline{\mathcal{D}}\times \mathbb{R}^{m})$ 
such that
$\frac{\partial H}{\partial u_{k}}=f_{k}$,
$k=1,\dots ,m $.

Submitted April 10, 2008. Published January 02, 2009.
Math Subject Classifications: 35J45, 35J55.
Key Words: Semi linear systems; Pohozaev identity; trivial solution;
           Robin boundary condition.