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A priori estimates for elliptic problems via Liouville type theorems

  • * Corresponding author: Roberta Filippucci

    * Corresponding author: Roberta Filippucci

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

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  • 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.

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


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  • [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. RădulescuSingular 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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