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May  2010, 9(3): 731-740. doi: 10.3934/cpaa.2010.9.731

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

Sándor Kelemen 1, and  Pavol Quittner 2,

1. 

Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 84173 Bratislava, Slovak Republic
2. 

Institute of Applied Mathematics and Statistics, Comenius University, Mlynská dolina, 84248 Bratislava

Received  June 2009 Revised  November 2009 Published  January 2010

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.
Keywords: critical exponent, a-priori bounds., Very weak solution, elliptic system.
Mathematics Subject Classification: Primary: 35J55, 35J65; Secondary: 35B33, 35B45, 35B6.
Citation: Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731
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