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J. Korean Math. Soc. 2020; 57(6): 1573-1590

Published online November 1, 2020 https://doi.org/10.4134/JKMS.j190836

Copyright © The Korean Mathematical Society.

The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders

Lidan Wang, Lihe Wang, Chunqin Zhou

Shanghai Jiao Tong University; Shanghai Jiao Tong University; Shanghai Jiao Tong University

Abstract

In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of $Lu(\mathbf{x})=a_{ij}(\mathbf{x})D_{ij}u(\mathbf{x})+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ or $Lu(\mathbf{x})=D_{i}(a_{ij}(\mathbf{x})$ $D_{j}u(\mathbf{x}))+b_{i}(\mathbf{x})D_{i}u(\mathbf{x})+c(\mathbf{x})u(\mathbf{x})=f(\mathbf{x})$ in unbounded cylinders. After establishing that the Aleksandrov maximum principle and boundary Harnack inequality hold for bounded solutions, we show that all solutions bounded from below are linear combinations of solutions, which are sums of two special solutions that exponential growth at one end and exponential decay at the another end, and a bounded solution that corresponds to the inhomogeneous term $f$ of the equation.

Keywords: Unbounded cylinder, Aleksandrov maximum principle, boundary Harnack inequality

MSC numbers: 35J25

Supported by: This work was financially supported by NSFC of China, No. 11771285

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