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, 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. AmendolaL. 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. BaoL. 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. CaffarelliM. G. CrandallM. 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. CrandallH. 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-DolcettaF. 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. WangL. 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

show all references

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. AmendolaL. 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. BaoL. 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. CaffarelliM. G. CrandallM. 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. CrandallH. 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-DolcettaF. 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. WangL. 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

[1]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[2]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[3]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[4]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[5]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

[6]

Irena PawŃow, Wojciech M. Zajączkowski. Global regular solutions to three-dimensional thermo-visco-elasticity with nonlinear temperature-dependent specific heat. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1331-1372. doi: 10.3934/cpaa.2017065

[7]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[8]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[9]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[10]

F.J. Herranz, J. de Lucas, C. Sardón. Jacobi--Lie systems: Fundamentals and low-dimensional classification. Conference Publications, 2015, 2015 (special) : 605-614. doi: 10.3934/proc.2015.0605

[11]

Mats Gyllenberg, Jifa Jiang, Lei Niu, Ping Yan. On the classification of generalized competitive Atkinson-Allen models via the dynamics on the boundary of the carrying simplex. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 615-650. doi: 10.3934/dcds.2018027

[12]

Deren Han, Zehui Jia, Yongzhong Song, David Z. W. Wang. An efficient projection method for nonlinear inverse problems with sparsity constraints. Inverse Problems & Imaging, 2016, 10 (3) : 689-709. doi: 10.3934/ipi.2016017

[13]

Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933

[14]

Guido De Philippis, Antonio De Rosa, Jonas Hirsch. The area blow up set for bounded mean curvature submanifolds with respect to elliptic surface energy functionals. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7031-7056. doi: 10.3934/dcds.2019243

[15]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[16]

Giovanni Cimatti. Forced periodic solutions for piezoelectric crystals. Communications on Pure & Applied Analysis, 2005, 4 (2) : 475-485. doi: 10.3934/cpaa.2005.4.475

[17]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[18]

Ian Schindler, Kyril Tintarev. Mountain pass solutions to semilinear problems with critical nonlinearity. Conference Publications, 2007, 2007 (Special) : 912-919. doi: 10.3934/proc.2007.2007.912

[19]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[20]

Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817

2019 Impact Factor: 1.105

Article outline

[Back to Top]