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A note on coupled nonlinear Schrödinger systems under the effect of general nonlinearities
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Exponential stability in linear viscoelasticity with almost flat memory kernels
Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions
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 |
  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.
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