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In this talk, we will introduce the properties of positive solutions to some kinds of the elliptic equation in unbounded cylinder with zero boundary condition. After establishing an Aleksandrov-Bakelman-Pucci maximum principle in unbounded cylinder, the solution spaces can be classified. In particular, we will show that one solution space where the solution is exponential growth at one end is one

dimensional, the other solution space where the solution is exponential decay at the another end is also one dimensional. These are some joint work with Lidan Wang and Lihe Wang.

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?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????ぷ??表??Ann. Inst. H. Poincar? Anal. Non Lin?aire, Calc. Var. Partial Differential Equations, Comm. Partial Differential Equations, J. Differential Geom., J. Funct. Anal., J. Math. Pures Appl., Math. Ann. ??????????