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Abstract
Consider the elliptic system
$-\Delta u=f(x,v)$, $-\Delta v+v=g(x,u)$
in a bounded smooth domain $\Omega\subset\R^N$,
complemented by the boundary conditions
$u=\partial_\nu v = 0$ on $\partial\Omega$.
Here $f,g$ are nonnegative Carathéodory functions
satisfying the growth conditions
$f\leq C(1+|v|^p)$, $g\leq C(1+|u|^q)$.
We find necessary and sufficient conditions
on $p,q$ guaranteeing that
$u,v\in L^\infty(\Omega)$
for any very weak solution $(u,v)$.
In addition, our conditions guarantee
the a priori estimate
$||u||_\infty+||v||_\infty\leq C$,
where $C$ depends only on the norm of $(u,v)$ in
$L^1_\delta(\Omega)\times L^1(\Omega)$.
 
Let us consider the borderline
in the $(p,q)$-plane between the region
where all very weak solutions are bounded
and the region where
unbounded solutions exist.
It turns out that this borderline
coincides with the corresponding borderline
for the system with the Neumann boundary
conditions $\partial_\nu u=\partial_\nu v = 0$
on $\partial\Omega$
if $p\leq N/(N-2)$,
while it coincides with the borderline
for the system with the Dirichlet
boundary conditions $u=v=0$ on $\partial\Omega$
if $p\geq(N+1)/(N-2)$.
If $p\in (N/(N-2),(N+1)/(N-2))$
then the borderline for the Dirichlet-Neumann problem
lies strictly between
the borderlines for the systems with pure Neumann
and pure Dirichlet boundary conditions.
 
Our proofs are based on some new
$L^p-L^q$ estimates in weighted $L^p$-spaces.
Mathematics Subject Classification: Primary: 35J55, 35J65; Secondary: 35B33, 35B45, 35B65.
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