We investigate the qualitative properties of solutions to the Zaremba type problem in unbounded domains for non-divergence elliptic equation with possible degeneration at infinity. The main result is a Phragmén–Lindelöf type principle on growth/decay of a solution at infinity depending on both the structure of the Neumann portion of the boundary and the “thickness” of its Dirichlet portion. The result is formulated in terms of the so-called s-capacity of the Dirichlet portion of the boundary, while the Neumann boundary should satisfy certain “admissibility” condition in the sequence of layers converging to infinity.
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