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Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities
The Dirichlet problem for fully nonlinear elliptic equations nondegenerate in a fixed direction
1.  Dipartimento di Matematica Pura e Applicata, Università degli Studi di Padova, Via Belzoni, 7, 35131, Padova 
References:
[1] 
M. Bardi and S. Bottacin, On the Dirichlet problem for nonlinear degenerate elliptic equations and applications to optimal control, Rend. Sem. Mat. Univ. Pol. Torino, 56 (1998), 1339. 
[2] 
M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of HamiltonJacobi Bellman Equations," Systems and Control: Foundations and Applications. Birkhauser, Boston, MA, 1997. 
[3] 
M. Bardi and P. Mannucci, On the Dirichlet problem for nontotally degenerate fully nonlinear elliptic equations, Commun. Pure Appl. Anal., 5 (2006), 709731. 
[4] 
M. Bardi and P. Mannucci, Comparison principles for subelliptic equations of MongeAmpère type, Boll. Unione Mat. Ital., 9 (2008), 489495. 
[5] 
M. Bardi and P. Mannucci, Comparison principles for equations of MongeAmpère type in Carnot groups: a direct proof, Lecture Notes of Seminario Interdisciplinare di Matematica, 7 (2008), 4151. 
[6] 
M. Bardi and P. Mannucci, Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of MongeAmpère type, to appear on Forum Math., published online May 2013, 20130067. doi: DOI: 10.1515/forum20130067. 
[7] 
F. H. Beatrous, T. J. Bieske and J. J. Manfredi, The maximum principle for vector fields, in "The $p$harmonic Equation and Recent Advances in Analysis," Contemp. Math., 370, 
[8] 
T. Bieske, On infinite harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002), 727761. 
[9] 
T. Bieske, Viscosity solutions on Grushintype planes, Illinois J. Math., 46 (2002), 893911. 
[10] 
T. Bieske and L. Capogna, The AronssonEuler equation for absolutely minimizing Lipschitz extensions with respect to CarnotCarathodory metrics, Trans. Amer. Math. Soc., 357 (2005), 795823. 
[11] 
I. Birindelli, I. Capuzzo Dolcetta and A. Cutrì, Indefinite semilinear equations on the Heisenberg group: a priori bounds and existence, Comm. Partial Differential Equations, 23 (1998), 11231157. 
[12] 
I. Birindelli and B. Stroffolini, Existence theorems for fully nonlinear equations in the Heisenberg group, Subelliptic PDE's and applications to geometry and finance, Lect. Notes Semin. Interdiscip. Mat., Potenza, 6 (2007), 4955. 
[13] 
A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory For Their SubLaplacians," Springer, Berlin, 2007. 
[14] 
M. G. Crandall, Viscosity solutions: a primer, In "Viscosity Solutions and Applications," Lecture Notes in Mathematics, 1660. Springer, Berlin; C.I.M.E., Florence, 1997. 
[15] 
M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of secondorder partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 167. 
[16] 
A. Cutrì and N. Tchou, Barrier functions for PucciHeisenberg operators and applications, Int. J. Dyn. Syst. Differ. Equ., 1, 2 (2007), 117131. 
[17] 
G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups," Princeton University Press, Princeton, 1982. 
[18] 
C. E. Gutierrez and A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, Comm. Partial Differential Equations, 29 (2004), 13051334. 
[19] 
L. Hörmander, Hypoelliptic Second Order Differential Equations, Acta Math. Uppsala, 119 (1967), 14717 . 
[20] 
M. A. Katsoulakis, A representation formula and regularizing properties for viscosity solutions of secondorder fully nonlinear degenerate parabolic equations, Nonlinear Analysis, Theory, Methods & Appl., 24 (1995), 147158. 
[21] 
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear secondorder elliptic partial differential equations, J. Diff. Eq., 83 (1990), 2678. 
[22] 
R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech., 101 (1988), 127. 
[23] 
J. J. Manfredi, Nonlinear subelliptic equations on Carnot groups, Notes of a course given at the Third School on Analysis and Geometry in Metric Spaces, Trento, May 2003, available at http://www.pitt.edu/ manfredi/. 
[24] 
C. Y. Wang, The Aronsson equation for absolute minimizers of $L^\infty$functionals associated with vector fields satisfying Hörmander's condition, Trans. Amer. Math. Soc. 359, 1 (2007), 91113. 
show all references
References:
[1] 
M. Bardi and S. Bottacin, On the Dirichlet problem for nonlinear degenerate elliptic equations and applications to optimal control, Rend. Sem. Mat. Univ. Pol. Torino, 56 (1998), 1339. 
[2] 
M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of HamiltonJacobi Bellman Equations," Systems and Control: Foundations and Applications. Birkhauser, Boston, MA, 1997. 
[3] 
M. Bardi and P. Mannucci, On the Dirichlet problem for nontotally degenerate fully nonlinear elliptic equations, Commun. Pure Appl. Anal., 5 (2006), 709731. 
[4] 
M. Bardi and P. Mannucci, Comparison principles for subelliptic equations of MongeAmpère type, Boll. Unione Mat. Ital., 9 (2008), 489495. 
[5] 
M. Bardi and P. Mannucci, Comparison principles for equations of MongeAmpère type in Carnot groups: a direct proof, Lecture Notes of Seminario Interdisciplinare di Matematica, 7 (2008), 4151. 
[6] 
M. Bardi and P. Mannucci, Comparison principles and Dirichlet problem for fully nonlinear degenerate equations of MongeAmpère type, to appear on Forum Math., published online May 2013, 20130067. doi: DOI: 10.1515/forum20130067. 
[7] 
F. H. Beatrous, T. J. Bieske and J. J. Manfredi, The maximum principle for vector fields, in "The $p$harmonic Equation and Recent Advances in Analysis," Contemp. Math., 370, 
[8] 
T. Bieske, On infinite harmonic functions on the Heisenberg group, Comm. Partial Differential Equations, 27 (2002), 727761. 
[9] 
T. Bieske, Viscosity solutions on Grushintype planes, Illinois J. Math., 46 (2002), 893911. 
[10] 
T. Bieske and L. Capogna, The AronssonEuler equation for absolutely minimizing Lipschitz extensions with respect to CarnotCarathodory metrics, Trans. Amer. Math. Soc., 357 (2005), 795823. 
[11] 
I. Birindelli, I. Capuzzo Dolcetta and A. Cutrì, Indefinite semilinear equations on the Heisenberg group: a priori bounds and existence, Comm. Partial Differential Equations, 23 (1998), 11231157. 
[12] 
I. Birindelli and B. Stroffolini, Existence theorems for fully nonlinear equations in the Heisenberg group, Subelliptic PDE's and applications to geometry and finance, Lect. Notes Semin. Interdiscip. Mat., Potenza, 6 (2007), 4955. 
[13] 
A. Bonfiglioli, E. Lanconelli and F. Uguzzoni, "Stratified Lie Groups and Potential Theory For Their SubLaplacians," Springer, Berlin, 2007. 
[14] 
M. G. Crandall, Viscosity solutions: a primer, In "Viscosity Solutions and Applications," Lecture Notes in Mathematics, 1660. Springer, Berlin; C.I.M.E., Florence, 1997. 
[15] 
M. G. Crandall, H. Ishii and P. L. Lions, User's guide to viscosity solutions of secondorder partial differential equations, Bull. Amer. Math. Soc., 27 (1992), 167. 
[16] 
A. Cutrì and N. Tchou, Barrier functions for PucciHeisenberg operators and applications, Int. J. Dyn. Syst. Differ. Equ., 1, 2 (2007), 117131. 
[17] 
G. B. Folland and E. M. Stein, "Hardy Spaces on Homogeneous Groups," Princeton University Press, Princeton, 1982. 
[18] 
C. E. Gutierrez and A. Montanari, Maximum and comparison principles for convex functions on the Heisenberg group, Comm. Partial Differential Equations, 29 (2004), 13051334. 
[19] 
L. Hörmander, Hypoelliptic Second Order Differential Equations, Acta Math. Uppsala, 119 (1967), 14717 . 
[20] 
M. A. Katsoulakis, A representation formula and regularizing properties for viscosity solutions of secondorder fully nonlinear degenerate parabolic equations, Nonlinear Analysis, Theory, Methods & Appl., 24 (1995), 147158. 
[21] 
H. Ishii and P. L. Lions, Viscosity solutions of fully nonlinear secondorder elliptic partial differential equations, J. Diff. Eq., 83 (1990), 2678. 
[22] 
R. Jensen, The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations, Arch. Rational Mech., 101 (1988), 127. 
[23] 
J. J. Manfredi, Nonlinear subelliptic equations on Carnot groups, Notes of a course given at the Third School on Analysis and Geometry in Metric Spaces, Trento, May 2003, available at http://www.pitt.edu/ manfredi/. 
[24] 
C. Y. Wang, The Aronsson equation for absolute minimizers of $L^\infty$functionals associated with vector fields satisfying Hörmander's condition, Trans. Amer. Math. Soc. 359, 1 (2007), 91113. 
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