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

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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

Abstract
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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 and 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.
[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.
[3]	L. A. Caffarelli and X. Cabrè, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	S. Koike, A Beginners Guide to the Theory of Viscosity Solutions, MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	A. Swiech, $W^{1,p}$-Interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.
[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.
[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.

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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.
[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.
[3]	L. A. Caffarelli and X. Cabrè, Fully Nonlinear Elliptic Equations, American Mathematical Society Colloquium Publications, 43, American Mathematical Society, Providence, RI, 1995.
[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.
[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.
[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.
[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.
[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.
[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.
[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.
[11]	S. Koike, A Beginners Guide to the Theory of Viscosity Solutions, MSJ Memoirs 13, Math. Soc. Japan, Tokyo, 2004.
[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.
[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.
[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.
[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.
[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.
[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.
[18]	A. Swiech, $W^{1,p}$-Interior estimates for solutions of fully nonlinear, uniformly elliptic equations, Adv. Differential Equations, 2 (1997), 1005-1027.
[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.
[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.

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