Nonexistence results for semilinear systems in unbounded domains

Khodja, Brahim ; Moussaoui, Abdelkrim

Electronic journal of differential equations, Tome 2009 (2009)

Résumé

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

$ 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 $.

Détail

Classification : 35J45, 35J55
Keywords: semi linear systems, pohozaev identity, trivial solution, Robin boundary condition