Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions

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.