Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders

Lidan Wang; Lihe Wang; Chunqin Zhou

doi:10.3934/cpaa.2021019

Article Contents

2021, Volume 20, Issue 3: 1241-1261. Doi: 10.3934/cpaa.2021019

This issue Previous Article Global well-posedness of the

$ n $

-dimensional hyper-dissipative Boussinesq system without thermal diffusivity Next Article Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation

Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders

1.
School of Mathematical Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
2.
Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA
3.
School of Mathematical Sciences, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, China

^* Corresponding author
^* Corresponding author

Received: August 2020

Revised: December 2020

Early access: February 2021

Published: March 2021

The third author is partially supported by NSFC Grant 11771285 and 12031012

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Abstract

In this paper, we consider the positive viscosity solutions for certain fully nonlinear uniformly elliptic equations in unbounded cylinder with zero boundary condition. After establishing an Aleksandrov-Bakelman-Pucci maximum principle, we classify all positive solutions as three categories in unbounded cylinder. Two special solution spaces (exponential growth at one end and exponential decay at the another) are one dimensional, independently, while solutions in the third solution space can be controlled by the solutions in the other two special solution spaces under some conditions, respectively.

Keywords:

Fully nonlinear elliptic equations,
unbounded cylinder,
Aleksandrov-Bakelman-Pucci maximum principle,
classification of solutions.

Mathematics Subject Classification: Primary: 35J60.

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