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March  2021, 20(3): 1241-1261. doi: 10.3934/cpaa.2021019

## 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

Received  August 2020 Revised  December 2020 Published  February 2021

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

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.

Citation: Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019
##### References:
 [1] S. Agmon, Lectures on exponential decay of solutions of second order elliptic equations: bounds on eigenfunctions of n-body schrodinger operations, Princeton, New Jersey: Princeton University Press, 1982.   Google Scholar [2] B. Avelin and V. Julin, A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term, J. Funct. Anal., 272 (2017), 3176 – 3215. doi: 10.1016/j.jfa.2016.12.026.  Google Scholar [3] M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second order elliptic equations in unbounded domains, Abstr. Appl. Anal., 2008 (2008), 1-20.  doi: 10.1155/2008/178534.  Google Scholar [4] J. Bao, L. Wang and C. Zhou, Positive solutions to elliptic equations in unbounded cylinder, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1389-1400.  doi: 10.3934/dcdsb.2016001.  Google Scholar [5] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, 1995. doi: 10.1090/coll/043.  Google Scholar [6] L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-398.  doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.3.CO;2-V.  Google Scholar [7] L. Cafarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981) 621–640. doi: 10.1512/iumj.1981.30.30049.  Google Scholar [8] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [9] I. Capuzzo-Dolcetta, F. Leoni and A. Vitolo, The Aleksandrof Backelman Pucci weak maximum priciple for Fully nonlinear equtions in unbounded domains, Commun. Partial Differ. Equ., 30 (2005), 1863-1881.  doi: 10.1080/03605300500300030.  Google Scholar [10] S. J. Gardiner, The Martin boundary of NTA strips, Bull. London Math. Soc., 22 (1990), 163-166.  doi: 10.1112/blms/22.2.163.  Google Scholar [11] M. Ghergu and J. Pres, Positive harmonic functions that vanish on a subset of a cylindrical surface, Potential Anal., 31 (2009), 147-181.  doi: 10.1007/s11118-009-9129-5.  Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer, Berlin, Heidelberg, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [13] E. M. Landis and N. S. Nadirashvili, Positive solutions of second-order equations in unbounded domains, Mat. Sb., 126 (1985), 133-139.   Google Scholar [14] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49 (1941), 137-172.  doi: 10.2307/1990054.  Google Scholar [15] M. G. Shur, The martin boundary for a linear, elliptic, second-order operator, Izv. Akad. Nauk. Ser. Mat., 27 (1963), 45-60.   Google Scholar [16] A. Swiech, W1, p interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differ. Equ., 2 (1997), 1005-1027.   Google Scholar [17] P. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order I. Lipschitz free boundaries are $C^{1, \alpha}$, Commun. Pure Appl. Math., 53 (2000), 799-810.  doi: 10.1002/(SICI)1097-0312(200007)53:7<799::AID-CPA1>3.0.CO;2-Q.  Google Scholar [18] L. Wang, L. Wang and C. Zhou, The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders, J. Korean Math. Soc., 57 (2020), 1573-1590.  doi: 10.4134/JKMS.j190836.  Google Scholar

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##### References:
 [1] S. Agmon, Lectures on exponential decay of solutions of second order elliptic equations: bounds on eigenfunctions of n-body schrodinger operations, Princeton, New Jersey: Princeton University Press, 1982.   Google Scholar [2] B. Avelin and V. Julin, A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term, J. Funct. Anal., 272 (2017), 3176 – 3215. doi: 10.1016/j.jfa.2016.12.026.  Google Scholar [3] M. E. Amendola, L. Rossi and A. Vitolo, Harnack inequalities and ABP estimates for nonlinear second order elliptic equations in unbounded domains, Abstr. Appl. Anal., 2008 (2008), 1-20.  doi: 10.1155/2008/178534.  Google Scholar [4] J. Bao, L. Wang and C. Zhou, Positive solutions to elliptic equations in unbounded cylinder, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1389-1400.  doi: 10.3934/dcdsb.2016001.  Google Scholar [5] L. Caffarelli and X. Cabré, Fully Nonlinear Elliptic Equations, American Mathematical Society, Providence, 1995. doi: 10.1090/coll/043.  Google Scholar [6] L. Caffarelli, M. G. Crandall, M. Kocan and A. Swiech, On viscosity solutions of fully nonlinear equations with measurable ingredients, Commun. Pure Appl. Math., 49 (1996), 365-398.  doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.3.CO;2-V.  Google Scholar [7] L. Cafarelli, E. Fabes, S. Mortola and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J., 30 (1981) 621–640. doi: 10.1512/iumj.1981.30.30049.  Google Scholar [8] M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 1-67.  doi: 10.1090/S0273-0979-1992-00266-5.  Google Scholar [9] I. Capuzzo-Dolcetta, F. Leoni and A. Vitolo, The Aleksandrof Backelman Pucci weak maximum priciple for Fully nonlinear equtions in unbounded domains, Commun. Partial Differ. Equ., 30 (2005), 1863-1881.  doi: 10.1080/03605300500300030.  Google Scholar [10] S. J. Gardiner, The Martin boundary of NTA strips, Bull. London Math. Soc., 22 (1990), 163-166.  doi: 10.1112/blms/22.2.163.  Google Scholar [11] M. Ghergu and J. Pres, Positive harmonic functions that vanish on a subset of a cylindrical surface, Potential Anal., 31 (2009), 147-181.  doi: 10.1007/s11118-009-9129-5.  Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Springer, Berlin, Heidelberg, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar [13] E. M. Landis and N. S. Nadirashvili, Positive solutions of second-order equations in unbounded domains, Mat. Sb., 126 (1985), 133-139.   Google Scholar [14] R. S. Martin, Minimal positive harmonic functions, Trans. Amer. Math. Soc., 49 (1941), 137-172.  doi: 10.2307/1990054.  Google Scholar [15] M. G. Shur, The martin boundary for a linear, elliptic, second-order operator, Izv. Akad. Nauk. Ser. Mat., 27 (1963), 45-60.   Google Scholar [16] A. Swiech, W1, p interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differ. Equ., 2 (1997), 1005-1027.   Google Scholar [17] P. Wang, Regularity of free boundaries of two-phase problems for fully nonlinear elliptic equations of second order I. Lipschitz free boundaries are $C^{1, \alpha}$, Commun. Pure Appl. Math., 53 (2000), 799-810.  doi: 10.1002/(SICI)1097-0312(200007)53:7<799::AID-CPA1>3.0.CO;2-Q.  Google Scholar [18] L. Wang, L. Wang and C. Zhou, The exponential growth and decay properties for solutions to elliptic equations in unbounded cylinders, J. Korean Math. Soc., 57 (2020), 1573-1590.  doi: 10.4134/JKMS.j190836.  Google Scholar
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