# American Institute of Mathematical Sciences

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

 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.
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). 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. doi: 10.2307/1971480. [3] L. A. Caffarelli and X. Cabrè, Fully Nonlinear Elliptic Equations,, American Mathematical Society Colloquium Publications, (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. 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. 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. 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. 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. 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. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd edition, (1983). doi: 10.1007/978-3-642-61798-0. [11] S. Koike, A Beginners Guide to the Theory of Viscosity Solutions,, MSJ Memoirs 13, 13 (2004). [12] S. Koike and A. Swiech, Maximum principle for fully nonlinear equations via the iterated comparison function method,, Math. Ann., 339 (2007), 461. 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. [14] E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen,, Proc. London Math. Soc., 13 (1913), 43. 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. 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, (2010), 193. [17] V. G. Maz'ya and T. O. Shaposhnikova, Sharp pointwise interpolation inequalities for derivatives,, Funct. Anal. Appl., 36 (2002), 30. 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. [19] A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains,, J. Differential Equations, 194 (2003), 166. 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. 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). 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. doi: 10.2307/1971480. [3] L. A. Caffarelli and X. Cabrè, Fully Nonlinear Elliptic Equations,, American Mathematical Society Colloquium Publications, (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. 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. 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. 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. 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. 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. [10] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order,, 2nd edition, (1983). doi: 10.1007/978-3-642-61798-0. [11] S. Koike, A Beginners Guide to the Theory of Viscosity Solutions,, MSJ Memoirs 13, 13 (2004). [12] S. Koike and A. Swiech, Maximum principle for fully nonlinear equations via the iterated comparison function method,, Math. Ann., 339 (2007), 461. 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. [14] E. Landau, Einige Ungleichungen für zweimal differenzierbare Funktionen,, Proc. London Math. Soc., 13 (1913), 43. 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. 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, (2010), 193. [17] V. G. Maz'ya and T. O. Shaposhnikova, Sharp pointwise interpolation inequalities for derivatives,, Funct. Anal. Appl., 36 (2002), 30. 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. [19] A. Vitolo, On the maximum principle for complete second-order elliptic operators in general domains,, J. Differential Equations, 194 (2003), 166. 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. doi: 10.1016/j.jmaa.2004.04.067.
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