Electron. J. Diff. Eqns., Vol. 2009(2009), No. 02, pp. 1-11.

Nonexistence results for semilinear systems in unbounded domains

Brahim Khodja, Abdelkrim Moussaoui

Abstract:
This paper concerns the non-existence of nontrivial solutions for the semi-linear system of gradient type
$$\displaylines{ \lambda \frac{\partial ^{2}u_{k}}{\partial t^{2}} -\sum_{i=1}^n \frac{\partial }{\partial x_{i}}(p_{i}(x)\frac{ \partial u_{k}}{\partial x_{i}})+f_{k}(x,u_{1},\dots ,u_{m}) =0\quad \hbox{in }\Omega ,\; k=1,\dots ,m }$$
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
$$ 2H(x,u_{1},\dots ,u_{m})-\sum_{k=1}^m u_{k}f_{k}(x,u_{1},\dots ,u_{m})\geq 0\quad (\hbox{resp.}\leq 0) $$
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 2, 2009.
Math Subject Classifications: 35J45, 35J55.
Key Words: Semi linear systems; Pohozaev identity; trivial solution; Robin boundary condition.

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Brahim Khodja
Department of mathematics, Badji Mokhtar University
B.P. 12 Annaba, Algeria
email: bmkhodja@yahoo.fr
Abdelkrim Moussaoui
Department of mathematics, Badji Mokhtar University
B.P. 12 Annaba, Algeria
email: remdz@yahoo.fr

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