A priori estimates for elliptic problems via Liouville type theorems

Previous Article
Evolution fractional differential problems with impulses and nonlocal conditions
DCDS-S Home
This Issue
Next Article
Preface to this special issue

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

A priori estimates for elliptic problems via Liouville type theorems

Laura Baldelli ^1, and Roberta Filippucci ^2,,

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.

Keywords: A priori estimates, elliptic problems, $ p $-$ q $-Laplacian systems.

Mathematics Subject Classification: Primary: 35J92, 35J70; Secondary: 35J47.

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-Artzi, P. 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. Clément, J. Fleckinger, E. 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. Clément, R. 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. Filippucci, P. 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. Poláčik, P. 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-Artzi, P. 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. Clément, J. Fleckinger, E. 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. Clément, R. 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. Filippucci, P. 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. Poláčik, P. 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.

[1]	Jianguo Huang, Jun Zou. Uniform a priori estimates for elliptic and static Maxwell interface problems. Discrete and Continuous Dynamical Systems - B, 2007, 7 (1) : 145-170. doi: 10.3934/dcdsb.2007.7.145
[2]	Théophile Chaumont-Frelet, Serge Nicaise, Jérôme Tomezyk. Uniform a priori estimates for elliptic problems with impedance boundary conditions. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2445-2471. doi: 10.3934/cpaa.2020107
[3]	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
[4]	Dimitri Mugnai, Kanishka Perera, Edoardo Proietti Lippi. A priori estimates for the Fractional p-Laplacian with nonlocal Neumann boundary conditions and applications. Communications on Pure and Applied Analysis, 2022, 21 (1) : 275-292. doi: 10.3934/cpaa.2021177
[5]	Alfonso Castro, Rosa Pardo. A priori estimates for positive solutions to subcritical elliptic problems in a class of non-convex regions. Discrete and Continuous Dynamical Systems - B, 2017, 22 (3) : 783-790. doi: 10.3934/dcdsb.2017038
[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]	Sándor Kelemen, Pavol Quittner. Boundedness and a priori estimates of solutions to elliptic systems with Dirichlet-Neumann boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (3) : 731-740. doi: 10.3934/cpaa.2010.9.731
[8]	Bojing Shi. $ W^{1, p} $ estimates for elliptic problems with drift terms in Lipschitz domains. Discrete and Continuous Dynamical Systems, 2022, 42 (2) : 537-553. doi: 10.3934/dcds.2021127
[9]	Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1121-1147. doi: 10.3934/dcdsb.2021083
[10]	Samer Dweik. $ L^{p, q} $ estimates on the transport density. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134
[11]	D. Bartolucci, L. Orsina. Uniformly elliptic Liouville type equations: concentration compactness and a priori estimates. Communications on Pure and Applied Analysis, 2005, 4 (3) : 499-522. doi: 10.3934/cpaa.2005.4.499
[12]	Pavol Quittner, Philippe Souplet. A priori estimates of global solutions of superlinear parabolic problems without variational structure. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1277-1292. doi: 10.3934/dcds.2003.9.1277
[13]	Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065
[14]	Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations and Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271
[15]	Dian Palagachev, Lubomira Softova. A priori estimates and precise regularity for parabolic systems with discontinuous data. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 721-742. doi: 10.3934/dcds.2005.13.721
[16]	L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure and Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9
[17]	Gabriele Bonanno, Giuseppina D'Aguì. Mixed elliptic problems involving the $p-$Laplacian with nonhomogeneous boundary conditions. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5797-5817. doi: 10.3934/dcds.2017252
[18]	Dung Le. On the regular set of BMO weak solutions to $p$-Laplacian strongly coupled nonregular elliptic systems. Discrete and Continuous Dynamical Systems - B, 2014, 19 (10) : 3245-3265. doi: 10.3934/dcdsb.2014.19.3245
[19]	Caiyan Li, Dongsheng Li. $ W^{1,p} $ estimates for elliptic systems on composite material with almost partially BMO coefficients. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3143-3159. doi: 10.3934/cpaa.2021100
[20]	Claudianor O. Alves, Vincenzo Ambrosio, Teresa Isernia. Existence, multiplicity and concentration for a class of fractional $ p \& q $ Laplacian problems in $ \mathbb{R} ^{N} $. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2009-2045. doi: 10.3934/cpaa.2019091

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Laura Baldelli Roberta Filippucci
on Google Scholar
Laura Baldelli Roberta Filippucci

2021 Impact Factor: 1.865

Tools

Article outline

2021 Impact Factor: 1.865

Tools

Metrics

Other articles
by authors

on AIMS
Laura Baldelli Roberta Filippucci
on Google Scholar
Laura Baldelli Roberta Filippucci

[Back to Top]