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This work deals with the system $(-\Delta)^m u= a(x) v^p$, $(-\Delta)^mv=b(x) u^q$ with Dirichlet boundary condition in a domain $\Omega\subset\RR^n$,where $\Omega$ is a ball if $n\ge 3$ or a smooth perturbation of a ball when$n=2$.  We prove that, under appropriate conditions on the parameters($a,b,p,q,m,n$), any non-negative solution $(u,v)$ of the system is bounded bya constant independent of $(u,v)$. Moreover, we prove that the conditions aresharp in the sense that, up to some border case, the relation on the parametersare also necessary.  The case $m=1$ was considered by Souplet in \cite{PS}. Our paper generalizeto $m\ge 1$ the results of that paper.

On the existence of bounded solutions for a nonlinear elliptic system