Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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THE EXPONENTIAL GROWTH AND DECAY PROPERTIES FOR SOLUTIONS TO ELLIPTIC EQUATIONS IN UNBOUNDED CYLINDERS

  • Wang, Lidan (School of Mathematical Sciences Shanghai Jiao Tong University) ;
  • Wang, Lihe (Department of Mathematics University of Iowa) ;
  • Zhou, Chunqin (School of Mathematical Sciences Shanghai Jiao Tong University)
  • Received : 2019.12.12
  • Accepted : 2020.03.25
  • Published : 2020.11.01
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Abstract

In this paper, we classify all solutions bounded from below to uniformly elliptic equations of second order in the form of Lu(x) = aij(x)Diju(x) + bi(x)Diu(x) + c(x)u(x) = f(x) or Lu(x) = Di(aij(x)Dju(x)) + bi(x)Diu(x) + c(x)u(x) = f(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.

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References

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