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December  2014, 7(6): 1335-1346. doi: 10.3934/dcdss.2014.7.1335

On the growth of positive entire solutions of elliptic PDEs and their gradients

Antonio Vitolo 1,

1. 

Department of Mathematics, University of Salerno, via Giovanni Paolo II n.132, 84084 Fisciano (SA), Italy

Received  January 2013 Revised  July 2013 Published  June 2014

We investigate the growth of entire positive functions $u(x)$ and their gradients $Du$ in Sobolev spaces when a polynomial growth is assumed for their image $Lu$ through a linear second-order uniform elliptic operator $L$. In particular, under suitable assumptions on the coefficients, we show that if $Lu$ is bounded, then $u(x)$ may grow at most quadratically at infinity. We also discuss, by counterexamples, the optimality of the assumptions and extend the results to viscosity solutions of fully nonlinear equations.
Keywords: gradient estimates, Entire solutions, elliptic equations, positive solutions, viscosity solutions..
Mathematics Subject Classification: Primary: 35B08, 35B09; Secondary: 35J6.
Citation: Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335
References:
[1]

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), Art. ID 178534, 19 pp. doi: 10.1155/2008/178534.  Google Scholar

[2]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar

[3]

L. A. Caffarelli and X. Cabrè, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[4]

L. A. 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), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A.  Google Scholar

[5]

I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, The Alexandrov-Bakelman-Pucci weak maximum principle for fully nonlinear equations in unbounded domains, Commun. Partial Differ. Equations, 30 (2005), 1863-1881. doi: 10.1080/03605300500300030.  Google Scholar

[6]

I. Capuzzo Dolcetta and A. Vitolo, A qualitative Phragmèn-Lindelöf theorem for fully nonlinear elliptic equations, J. Differential Equations, 243 (2007), 578-592. doi: 10.1016/j.jde.2007.08.001.  Google Scholar

[7]

I. Capuzzo Dolcetta and A. Vitolo, Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst., 28 (2010), 539-557. doi: 10.3934/dcds.2010.28.539.  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]

M. G. Crandall, M. Kocan, P. L. Lions and A. Swiech, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 24 (1999), 1-22.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin-New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[11]

S. Koike, A Beginners Guide to the Theory of Viscosity Solutions, MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004.  Google Scholar

[12]

S. Koike and A. Swiech, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339 (2007), 461-484. doi: 10.1007/s00208-007-0125-z.  Google Scholar

[13]

S. Koike and T. Takahashi, Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients, Adv. Differential Equations, 7 (2002), 493-512.  Google Scholar

[14]

E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen, Proc. London Math. Soc., 13 (1913), 43-49. doi: 10.1112/plms/s2-13.1.43.  Google Scholar

[15]

Y. Y. Li and L. Nirenberg, Generalization of a well-known inequality, Progress in Nonlinear Differential Equations and Their Applications, 66 (2006), 365-370. doi: 10.1007/3-7643-7401-2_24.  Google Scholar

[16]

Y. Y. Li and L. Nirenberg, A miscellany, in Percorsi incrociati (in ricordo di Vittorio Cafagna) (eds I. Capuzzo Dolcetta, M. Transirico and A. Vitolo), Collana Scientifica di Ateneo, Università di Salerno, 2010, 193-208. Google Scholar

[17]

V. G. Maz'ya and T. O. Shaposhnikova, Sharp pointwise interpolation inequalities for derivatives, Funct. Anal. Appl., 36 (2002), 30-48. doi: 10.1023/A:1014478100799.  Google Scholar

[18]

A. Swiech, $W^{1,p}$-Interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.  Google Scholar

[19]

A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains, J. Differential Equations, 194 (2003), 166-184. doi: 10.1016/S0022-0396(03)00193-1.  Google Scholar

[20]

A. Vitolo, On the Phragmèn-Lindelöf principle for second-order elliptic equations, J. Math. Anal. Appl., 300 (2004), 244-259. doi: 10.1016/j.jmaa.2004.04.067.  Google Scholar

show all references

References:
[1]

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), Art. ID 178534, 19 pp. doi: 10.1155/2008/178534.  Google Scholar

[2]

L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213. doi: 10.2307/1971480.  Google Scholar

[3]

L. A. Caffarelli and X. Cabrè, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.  Google Scholar

[4]

L. A. 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), 365-397. doi: 10.1002/(SICI)1097-0312(199604)49:4<365::AID-CPA3>3.0.CO;2-A.  Google Scholar

[5]

I. Capuzzo Dolcetta, F. Leoni and A. Vitolo, The Alexandrov-Bakelman-Pucci weak maximum principle for fully nonlinear equations in unbounded domains, Commun. Partial Differ. Equations, 30 (2005), 1863-1881. doi: 10.1080/03605300500300030.  Google Scholar

[6]

I. Capuzzo Dolcetta and A. Vitolo, A qualitative Phragmèn-Lindelöf theorem for fully nonlinear elliptic equations, J. Differential Equations, 243 (2007), 578-592. doi: 10.1016/j.jde.2007.08.001.  Google Scholar

[7]

I. Capuzzo Dolcetta and A. Vitolo, Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations, Discrete Contin. Dyn. Syst., 28 (2010), 539-557. doi: 10.3934/dcds.2010.28.539.  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]

M. G. Crandall, M. Kocan, P. L. Lions and A. Swiech, Existence results for boundary problems for uniformly elliptic and parabolic fully nonlinear equations, Electron. J. Differential Equations, 24 (1999), 1-22.  Google Scholar

[10]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd edition, Grundlehren der Mathematischen Wissenschaften, 224, Springer-Verlag, Berlin-New York, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[11]

S. Koike, A Beginners Guide to the Theory of Viscosity Solutions, MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004.  Google Scholar

[12]

S. Koike and A. Swiech, Maximum principle for fully nonlinear equations via the iterated comparison function method, Math. Ann., 339 (2007), 461-484. doi: 10.1007/s00208-007-0125-z.  Google Scholar

[13]

S. Koike and T. Takahashi, Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients, Adv. Differential Equations, 7 (2002), 493-512.  Google Scholar

[14]

E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen, Proc. London Math. Soc., 13 (1913), 43-49. doi: 10.1112/plms/s2-13.1.43.  Google Scholar

[15]

Y. Y. Li and L. Nirenberg, Generalization of a well-known inequality, Progress in Nonlinear Differential Equations and Their Applications, 66 (2006), 365-370. doi: 10.1007/3-7643-7401-2_24.  Google Scholar

[16]

Y. Y. Li and L. Nirenberg, A miscellany, in Percorsi incrociati (in ricordo di Vittorio Cafagna) (eds I. Capuzzo Dolcetta, M. Transirico and A. Vitolo), Collana Scientifica di Ateneo, Università di Salerno, 2010, 193-208. Google Scholar

[17]

V. G. Maz'ya and T. O. Shaposhnikova, Sharp pointwise interpolation inequalities for derivatives, Funct. Anal. Appl., 36 (2002), 30-48. doi: 10.1023/A:1014478100799.  Google Scholar

[18]

A. Swiech, $W^{1,p}$-Interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.  Google Scholar

[19]

A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains, J. Differential Equations, 194 (2003), 166-184. doi: 10.1016/S0022-0396(03)00193-1.  Google Scholar

[20]

A. Vitolo, On the Phragmèn-Lindelöf principle for second-order elliptic equations, J. Math. Anal. Appl., 300 (2004), 244-259. doi: 10.1016/j.jmaa.2004.04.067.  Google Scholar

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