May  2015, 14(3): 897-922. doi: 10.3934/cpaa.2015.14.897

Gradient estimates and comparison principle for some nonlinear elliptic equations

1. 

Università degli Studi di Napoli "Parthenope", Dipartimento di Ingegneria, Centro Direzionale, Isola C4 80143 Napoli, Italy

2. 

Università degli Studi di Napoli Federico II, Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Complesso Monte S. Angelo, Via Cintia, 80126 Napoli, Italy, Italy

3. 

Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Universitá di Napoli "Federico II", via Cintia, I-80126 Napoli

Received  July 2014 Revised  January 2015 Published  March 2015

We consider a class of Dirichlet boundary problems for nonlinear elliptic equations with a first order term. We show how the summability of the gradient of a solution increases when the summability of the datum increases. We also prove comparison principle which gives in turn uniqueness results by strenghtening the assumptions on the operators.
Citation: Maria Francesca Betta, Rosaria Di Nardo, Anna Mercaldo, Adamaria Perrotta. Gradient estimates and comparison principle for some nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (3) : 897-922. doi: 10.3934/cpaa.2015.14.897
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show all references

References:
[1]

Boll. Accademia Gioenia di Scienze Naturali in Catania, 46 (2013), 2-11. Google Scholar

[2]

J. Differential Equations, 12 (2010), 3279-3290. doi: 10.1016/j.jde.2010.07.030.  Google Scholar

[3]

Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1017-1054. doi: 10.1016/j.anihpc.2010.01.010.  Google Scholar

[4]

A. Alvino, V. Ferone and A. Mercaldo, Sharp a-priori estimates for a class of nonlinear elliptic equations with lower order terms,, Ann. Mat. Pura Appl., (): 10231.   Google Scholar

[5]

Ann. Mat. Pura Appl., 178 (2000), 129-142. doi: 10.1007/BF02505892.  Google Scholar

[6]

Mediterr. J. Math, 5 (2008), 173-185. doi: 10.1007/s00009-008-0142-5.  Google Scholar

[7]

Boll Unione Mat. Ital., 1 (2008), 645-662.  Google Scholar

[8]

Comm. Partial Differential Equations, 17 (1992), 1037-1050. doi: 10.1080/03605309208820876.  Google Scholar

[9]

Ann. Scuola Norm. Sup., Pisa Cl. Sci, 5 (2006), 107-136.  Google Scholar

[10]

Ann. Scuola Norm. Sup. Pisa Cl. Sci, 22 (1995), 241-273.  Google Scholar

[11]

Academic Press, Boston, 1988.  Google Scholar

[12]

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei Mat. Appl., 21 (2010), 1-14. doi: 10.4171/RLM/557.  Google Scholar

[13]

C. R. Math. Acad. Sci. Paris, 334 (2002), 757-762. doi: 10.1016/S1631-073X(02)02338-5.  Google Scholar

[14]

ESAIM Control Optim. Calc. Var. 8 (2002), 239-272. doi: 10.1051/cocv:2002051.  Google Scholar

[15]

Ricerche Mat., 63 (2014), 41-56. doi: 10.1007/s11587-014-0198-4.  Google Scholar

[16]

J. Eur. Math. Soc. (JEMS), 16 (2014), 571-595. doi: 10.4171/JEMS/440.  Google Scholar

[17]

Ann. Mat. Pura Appl., 170 (1996), 207-240. doi: 10.1007/BF01758989.  Google Scholar

[18]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 25 (1997), 375-396.  Google Scholar

[19]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 28 (1999), 741-808.  Google Scholar

[20]

Amer. J. Math., 133 (2011), 1093-1149. doi: 10.1353/ajm.2011.0023.  Google Scholar

[21]

Advanced Nonlinear Studies, 7 (2007), 31-46.  Google Scholar

[22]

J. Differential Equations, 256 (2014), 577-608. doi: 10.1016/j.jde.2013.09.013.  Google Scholar

[23]

Ann. Scuola Norm. Sup. Pisa Cl. Sci., 13 (2014), 137-205.  Google Scholar

[24]

Commun. Pure Appl. Anal., 7 (2008), 163-192.  Google Scholar

[25]

Enseignement Math., 12 (1966), 249-276.  Google Scholar

[26]

Lecture Notes in Mathematics, No. 1150 Springer, Berlin, 1985.  Google Scholar

[27]

Bull. Soc. Math. France, 93 (1965), 97-107.  Google Scholar

[28]

P.- L. Lions and F. Murat, Sur les solutions renormalisées d'equations elliptiques non linéaires,, manuscript., ().   Google Scholar

[29]

in Geometric Properties for Parabolics and Elliptic PDE's. Springer INdAM Series, Vol. 2 (2013), 223-235 doi: 10.1007/978-88-470-2841-8_14.  Google Scholar

[30]

Preprint 93023, Laboratoire d'Analyse Numérique de l'Université Paris VI (1993). Google Scholar

[31]

in On the notions of solution to nonlinear elliptic problems: results and developments, Quad. Mat. Dept. Math., Seconda Univ. di Napoli, Caserta, 23, (2008) 459-497.  Google Scholar

[32]

Ann. Mat. Pura Appl., 120 (1979), 160-184. doi: 10.1007/BF02411942.  Google Scholar

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