# American Institute of Mathematical Sciences

June  2010, 28(2): 539-557. doi: 10.3934/dcds.2010.28.539

## Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations

 1 Dipartimento di Matematica, Sapienza - Università di Roma, P.le A.Moro 2, Roma, I-00185, Italy 2 Dipartimento di Matematica e Informatica, Università di Salerno, P. Grahamstown, Fisciano, SA I-84084, Italy

Received  December 2009 Revised  April 2010 Published  April 2010

In this paper we discuss some extensions to a fully nonlinear setting of results by Y.Y. Li and L. Nirenberg [25] about gradient estimates for non-negative solutions of linear elliptic equations.Our approach relies heavily on methods developed by L. Caffarelli in [3] and [4].
Citation: Italo Capuzzo Dolcetta, Antonio Vitolo. Glaeser's type gradient estimates for non-negative solutions of fully nonlinear elliptic equations. Discrete and Continuous Dynamical Systems, 2010, 28 (2) : 539-557. doi: 10.3934/dcds.2010.28.539
 [1] Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete and Continuous Dynamical Systems, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343 [2] Francesca Da Lio. Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 395-415. doi: 10.3934/cpaa.2004.3.395 [3] Andrzej Świȩch. Pointwise properties of $L^p$-viscosity solutions of uniformly elliptic equations with quadratically growing gradient terms. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2945-2962. doi: 10.3934/dcds.2020156 [4] Shigeaki Koike, Andrzej Świech. Local maximum principle for $L^p$-viscosity solutions of fully nonlinear elliptic PDEs with unbounded coefficients. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1897-1910. doi: 10.3934/cpaa.2012.11.1897 [5] Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897 [6] Xavier Cabré, Manel Sanchón, Joel Spruck. A priori estimates for semistable solutions of semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 601-609. doi: 10.3934/dcds.2016.36.601 [7] Claudia Anedda, Giovanni Porru. Boundary estimates for solutions of weighted semilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2012, 32 (11) : 3801-3817. doi: 10.3934/dcds.2012.32.3801 [8] Zhijun Zhang. Large solutions of semilinear elliptic equations with a gradient term: existence and boundary behavior. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1381-1392. doi: 10.3934/cpaa.2013.12.1381 [9] Dumitru Motreanu, Viorica V. Motreanu, Abdelkrim Moussaoui. Location of Nodal solutions for quasilinear elliptic equations with gradient dependence. Discrete and Continuous Dynamical Systems - S, 2018, 11 (2) : 293-307. doi: 10.3934/dcdss.2018016 [10] Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure and Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465 [11] Ana Maria Bertone, J.V. Goncalves. Discontinuous elliptic problems in $R^N$: Lower and upper solutions and variational principles. Discrete and Continuous Dynamical Systems, 2000, 6 (2) : 315-328. doi: 10.3934/dcds.2000.6.315 [12] Y. Efendiev, Alexander Pankov. Meyers type estimates for approximate solutions of nonlinear elliptic equations and their applications. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 481-492. doi: 10.3934/dcdsb.2006.6.481 [13] Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54 [14] Tommaso Leonori, Ireneo Peral, Ana Primo, Fernando Soria. Basic estimates for solutions of a class of nonlocal elliptic and parabolic equations. Discrete and Continuous Dynamical Systems, 2015, 35 (12) : 6031-6068. doi: 10.3934/dcds.2015.35.6031 [15] Arunima Bhattacharya, Micah Warren. $C^{2, \alpha}$ estimates for solutions to almost Linear elliptic equations. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1363-1383. doi: 10.3934/cpaa.2021024 [16] A. Domoshnitsky. About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations. Conference Publications, 2011, 2011 (Special) : 373-380. doi: 10.3934/proc.2011.2011.373 [17] Cristian Enache. Maximum and minimum principles for a class of Monge-Ampère equations in the plane, with applications to surfaces of constant Gauss curvature. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1347-1359. doi: 10.3934/cpaa.2014.13.1347 [18] Giuseppe Riey. Regularity and weak comparison principles for double phase quasilinear elliptic equations. Discrete and Continuous Dynamical Systems, 2019, 39 (8) : 4863-4873. doi: 10.3934/dcds.2019198 [19] Massimiliano Berti, M. Matzeu, Enrico Valdinoci. On periodic elliptic equations with gradient dependence. Communications on Pure and Applied Analysis, 2008, 7 (3) : 601-615. doi: 10.3934/cpaa.2008.7.601 [20] Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128

2021 Impact Factor: 1.588