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 [
Citation: |
[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.![]() ![]() ![]() |