July  2020, 13(7): 1883-1898. doi: 10.3934/dcdss.2020148

A priori estimates for elliptic problems via Liouville type theorems

1. 

Department of Mathematics, University of Firenze, viale Morgagni 40-44, 50134 Firenze, Italy

2. 

Department of Mathematics, University of Perugia, via Vanvitelli 1, 06123 Perugia, Italy

* Corresponding author: Roberta Filippucci

Dedicated to Professor Patrizia Pucci on the occasion of her 65th birthday, with deep gratitude, esteem and affection

Received  November 2018 Revised  November 2018 Published  November 2019

In this paper we prove a priori estimates for positive solutions of elliptic equations of the $ p $-Laplacian type on arbitrary domains of $ \mathbb {R}^N $, when a nonlinearity depending on the gradient is considered. Also the case of systems with very general nonlinearities is considered. Our main theorems extend previous results by Polacik, Quitter and Souplet in [26] in which either the case $ p = 2 $ with a nonlinearity depending on the gradient or the $ p $-Laplacian case with a nonlinearity not depending on the gradient is treated. The technique is based on the use of a method developed in [26] whose main tools are rescaling arguments combined with a key "doubling" property, which is different from the celebrated blow up technique due to Gidas and Spruck in [16]. A discussion on the sharpness of the main result in the scalar case is presented.

Citation: Laura Baldelli, Roberta Filippucci. A priori estimates for elliptic problems via Liouville type theorems. Discrete and Continuous Dynamical Systems - S, 2020, 13 (7) : 1883-1898. doi: 10.3934/dcdss.2020148
References:
[1]

C. Azizieh and P. Clèment, A priori estimates and continuation methods for positive solutions of $p$-Laplace equations, J. Differential Equations, 179 (2002), 213-245.  doi: 10.1006/jdeq.2001.4029.

[2]

J.-P. Bartier, Global behavior of solutions of a reaction-diffusion equation with gradient absorption in unbounded domains, Asymptot. Anal., 46 (2006), 325-347. 

[3]

M. Ben-ArtziP. Souplet and F. B. Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl., 81 (2002), 343-378.  doi: 10.1016/S0021-7824(01)01243-0.

[4]

M. Chipot and F. B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.  doi: 10.1137/0520060.

[5]

P. ClémentJ. FleckingerE. Mitidieri and F. de Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations, 166 (2000), 455-477.  doi: 10.1006/jdeq.2000.3805.

[6]

P. ClémentR. Manásevich and E. Mitidieri, Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations, 18 (1993), 2071-2106.  doi: 10.1080/03605309308821005.

[7]

A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, Ⅱ, J. Differential Equations, 250 (2011), 4409-4436.  doi: 10.1016/j.jde.2011.02.016.

[8]

R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916.  doi: 10.1016/j.na.2008.12.018.

[9]

R. Filippucci, Nonexistence of nonnegative solutions of elliptic systems of divergence type, J. Diff. Equations, 250 (2011), 572-595.  doi: 10.1016/j.jde.2010.09.028.

[10]

R. Filippucci and C. Lini, Existence of solutions for quasilinear Dirichlet problems with gradient terms, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 267-286.  doi: 10.3934/dcdss.2019019.

[11]

R. FilippucciP. Pucci and M. Rigoli, Nonlinear weighted $p$-Laplacian elliptic inequalities with gradient terms, Comm. Cont. Math., 12 (2010), 501-535.  doi: 10.1142/S0219199710003841.

[12]

R. Filippucci and F. Vinti, Coercive elliptic systems with gradient terms, Advances in Nonlinear Analysis, 6 (2017), 165-182.  doi: 10.1515/anona-2016-0183.

[13]

M. Ghergu and V. Rădulescu, Nonradial blow-up solutions of sublinear elliptic equations with gradient terms, Comm. Pure Appl. An., 3 (2004), 465-474.  doi: 10.3934/cpaa.2004.3.465.

[14]

M. Ghergu and V. Rădulescu, On a class of sublinear elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646.  doi: 10.1016/j.jmaa.2005.03.012.

[15] M. Ghergu and V. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications, 37. The Clarendon Press, Oxford University Press, Oxford, 2008. 
[16]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[17]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[18]

H. A. Hamid and M. F. Bidaut-Véron, Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient, C. R. Math. Acad. Sci. Paris, 346 (2008), 1251-1256.  doi: 10.1016/j.crma.2008.10.002.

[19]

B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313. 

[20]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[21]

O. Martio and G. Porru, Large solutions of quasilinear elliptic equations in the degenerate case, Complex analysis and differential equations (Uppsala, 1997), Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., Uppsala Univ., Uppsala, 64 (1999), 225–241.

[22]

E. Mitidieri and S. I. Pohozaev, The absence of global positive solutions to quasilinear elliptic inequalities, Dokl. Akad. Nauk, 359 (1998), 456-460. 

[23]

E. Mitidieri and S. I. Pohozaev, Absence of positive solutions for a system of quasilinear elliptic equations and inequalities in $\mathbb {R}^N$, Dokl. Akad. Nauk, 366 (1999), 13-17. 

[24]

E. Mitidieri and S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 3 (2001), 1-362. 

[25]

W.-M. Ni and J. Serrin, Existence and non-existence theorems for ground states of quasilinear partial differential equations: The anomalous case, Atti Convegni Lincei, 77 (1986), 231-257. 

[26]

P. PoláčikP. Quitter and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I. Elliptic equations and systems, Duke Mathematical Journal, 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.

[27]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.

[28]

D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Diff. Equations, 199 (2004), 96-114.  doi: 10.1016/j.jde.2003.10.021.

[29]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.

[30]

P. Souplet, Finite time blowup for a nonlinear parabolic equation with a gradient term and applications, Math Methods Appl. Sci., 19 (1996), 1317-1333.  doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M.

[31]

P. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electron. J. Differential Equations, 10 (2001), 19 pp.

[32]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.

[33]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.

show all references

References:
[1]

C. Azizieh and P. Clèment, A priori estimates and continuation methods for positive solutions of $p$-Laplace equations, J. Differential Equations, 179 (2002), 213-245.  doi: 10.1006/jdeq.2001.4029.

[2]

J.-P. Bartier, Global behavior of solutions of a reaction-diffusion equation with gradient absorption in unbounded domains, Asymptot. Anal., 46 (2006), 325-347. 

[3]

M. Ben-ArtziP. Souplet and F. B. Weissler, The local theory for viscous Hamilton-Jacobi equations in Lebesgue spaces, J. Math. Pures Appl., 81 (2002), 343-378.  doi: 10.1016/S0021-7824(01)01243-0.

[4]

M. Chipot and F. B. Weissler, Some blow up results for a nonlinear parabolic problem with a gradient term, SIAM J. Math. Anal., 20 (1989), 886-907.  doi: 10.1137/0520060.

[5]

P. ClémentJ. FleckingerE. Mitidieri and F. de Thélin, Existence of positive solutions for a nonvariational quasilinear elliptic system, J. Differential Equations, 166 (2000), 455-477.  doi: 10.1006/jdeq.2000.3805.

[6]

P. ClémentR. Manásevich and E. Mitidieri, Positive solutions for a quasilinear system via blow up, Comm. Partial Differential Equations, 18 (1993), 2071-2106.  doi: 10.1080/03605309308821005.

[7]

A. Farina and J. Serrin, Entire solutions of completely coercive quasilinear elliptic equations, Ⅱ, J. Differential Equations, 250 (2011), 4409-4436.  doi: 10.1016/j.jde.2011.02.016.

[8]

R. Filippucci, Nonexistence of positive weak solutions of elliptic inequalities, Nonlinear Anal., 70 (2009), 2903-2916.  doi: 10.1016/j.na.2008.12.018.

[9]

R. Filippucci, Nonexistence of nonnegative solutions of elliptic systems of divergence type, J. Diff. Equations, 250 (2011), 572-595.  doi: 10.1016/j.jde.2010.09.028.

[10]

R. Filippucci and C. Lini, Existence of solutions for quasilinear Dirichlet problems with gradient terms, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 267-286.  doi: 10.3934/dcdss.2019019.

[11]

R. FilippucciP. Pucci and M. Rigoli, Nonlinear weighted $p$-Laplacian elliptic inequalities with gradient terms, Comm. Cont. Math., 12 (2010), 501-535.  doi: 10.1142/S0219199710003841.

[12]

R. Filippucci and F. Vinti, Coercive elliptic systems with gradient terms, Advances in Nonlinear Analysis, 6 (2017), 165-182.  doi: 10.1515/anona-2016-0183.

[13]

M. Ghergu and V. Rădulescu, Nonradial blow-up solutions of sublinear elliptic equations with gradient terms, Comm. Pure Appl. An., 3 (2004), 465-474.  doi: 10.3934/cpaa.2004.3.465.

[14]

M. Ghergu and V. Rădulescu, On a class of sublinear elliptic problems with convection term, J. Math. Anal. Appl., 311 (2005), 635-646.  doi: 10.1016/j.jmaa.2005.03.012.

[15] M. Ghergu and V. Rădulescu, Singular Elliptic Problems: Bifurcation and Asymptotic Analysis, Oxford Lecture Series in Mathematics and its Applications, 37. The Clarendon Press, Oxford University Press, Oxford, 2008. 
[16]

B. Gidas and J. Spruck, A priori bounds for positive solutions of nonlinear elliptic equations, Comm. Partial Differential Equations, 6 (1981), 883-901.  doi: 10.1080/03605308108820196.

[17]

B. Gidas and J. Spruck, Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math., 34 (1981), 525-598.  doi: 10.1002/cpa.3160340406.

[18]

H. A. Hamid and M. F. Bidaut-Véron, Correlation between two quasilinear elliptic problems with a source term involving the function or its gradient, C. R. Math. Acad. Sci. Paris, 346 (2008), 1251-1256.  doi: 10.1016/j.crma.2008.10.002.

[19]

B. Hu, Nonexistence of a positive solution of the Laplace equation with a nonlinear boundary condition, Differential Integral Equations, 7 (1994), 301-313. 

[20]

G. M. Lieberman, Boundary regularity for solutions of degenerate elliptic equations, Nonlinear Analysis: Theory, Methods & Applications, 12 (1988), 1203-1219.  doi: 10.1016/0362-546X(88)90053-3.

[21]

O. Martio and G. Porru, Large solutions of quasilinear elliptic equations in the degenerate case, Complex analysis and differential equations (Uppsala, 1997), Acta Univ. Upsaliensis Skr. Uppsala Univ. C Organ. Hist., Uppsala Univ., Uppsala, 64 (1999), 225–241.

[22]

E. Mitidieri and S. I. Pohozaev, The absence of global positive solutions to quasilinear elliptic inequalities, Dokl. Akad. Nauk, 359 (1998), 456-460. 

[23]

E. Mitidieri and S. I. Pohozaev, Absence of positive solutions for a system of quasilinear elliptic equations and inequalities in $\mathbb {R}^N$, Dokl. Akad. Nauk, 366 (1999), 13-17. 

[24]

E. Mitidieri and S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 3 (2001), 1-362. 

[25]

W.-M. Ni and J. Serrin, Existence and non-existence theorems for ground states of quasilinear partial differential equations: The anomalous case, Atti Convegni Lincei, 77 (1986), 231-257. 

[26]

P. PoláčikP. Quitter and P. Souplet, Singularity and decay estimates in superlinear problems via Liouville-type theorems, I. Elliptic equations and systems, Duke Mathematical Journal, 139 (2007), 555-579.  doi: 10.1215/S0012-7094-07-13935-8.

[27]

P. Pucci and J. Serrin, The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, 73. Birkhäuser Verlag, Basel, 2007.

[28]

D. Ruiz, A priori estimates and existence of positive solutions for strongly nonlinear problems, J. Diff. Equations, 199 (2004), 96-114.  doi: 10.1016/j.jde.2003.10.021.

[29]

J. Serrin and H. Zou, Cauchy-Liouville and universal boundness theorems for quasilinear elliptic equations and inequalities, Acta Math., 189 (2002), 79-142.  doi: 10.1007/BF02392645.

[30]

P. Souplet, Finite time blowup for a nonlinear parabolic equation with a gradient term and applications, Math Methods Appl. Sci., 19 (1996), 1317-1333.  doi: 10.1002/(SICI)1099-1476(19961110)19:16<1317::AID-MMA835>3.0.CO;2-M.

[31]

P. Souplet, Recent results and open problems on parabolic equations with gradient nonlinearities, Electron. J. Differential Equations, 10 (2001), 19 pp.

[32]

P. Tolksdorf, Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations, 51 (1984), 126-150.  doi: 10.1016/0022-0396(84)90105-0.

[33]

J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim., 12 (1984), 191-202.  doi: 10.1007/BF01449041.

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