Abstract
For a higher order quasilinear differential equation, the existence of uniform estimates for positive solutions with common domain of definition is proved; these estimates depend on the estimates for the coefficients of the equation and do not depend on the coefficients themselves.
Similar content being viewed by others
References
I. T. Kiguradze and T. A. Chanturiya, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations (Nauka, Moscow, 1990; Kluwer, Dordrecht, 1993).
I. V. Astashova, “Application of Dynamical Systems to the Study of Asymptotic Properties of Solutions to Nonlinear Higher-Order Differential Equations,” Sovrem. Mat. Prilozh. 8, 3–33 (2003) [J. Math. Sci. 126 (5), 1361–1391 (2005)].
T. A. Chanturiya, “Existence of Singular and Unbounded Oscillating Solutions of Differential Equations of Emden-Fowler Type,” Diff. Uravn. 28(6), 1009–1022 (1992) [Diff. Eqns. 28, 811–824 (1992)].
V. A. Kozlov, “On Kneser Solutions of Higher Order Nonlinear Ordinary Differential Equations,” Ark. Mat. 37(2), 305–322 (1999).
V. A. Kondrat’ev, “On Qualitative Properties of the Solutions of Semilinear Elliptic Equations,” Tr. Semin. im. I.G. Petrovskogo 16, 186–190 (1992) [J. Math. Sci. 69 (3), 1068–1071 (1996)].
G. G. Kvinkadze and I. T. Kiguradze, “On Rapidly Growing Solutions to Nonlinear Ordinary Differential Equations,” Soobshch. Akad. Nauk. Gruz. SSR 106(3), 465–468 (1982).
I. V. Astashova, “On Qualitative Properties of Solutions to Equations of Emden-Fowler Type,” Usp. Mat. Nauk 51(5), 185 (1996).
I. V. Astashova, “Estimates of Solutions to One-Dimensional Schrödinger Equation,” in Progress in Analysis: Proc. 3rd Int. ISAAC Congr. (World Sci., River Edge, NJ, 2003), Vol. 2, pp. 955–960.
G. Pólya, “On the Mean-Value Theorem Corresponding to a Given Linear Homogeneous Differential Equation,” Trans. Am. Math. Soc. 24, 312–324 (1924).
C. de La Vallée Poussin, “Sur l’équation différentielle linéaire du second ordre. Détermination d’une intégrale par deux valeurs assignées. Extension aux équations d’ordre n,” J. Math. Pures Appl. 8, 125–144 (1929).
A. Yu. Levin, “Non-oscillation of Solutions of the Equation x (n) + p1(t)x (n − 1) + ... + p n(t)x = 0,” Usp. Mat. Nauk 24(2), 43–96 (1969) [Russ. Math. Surv. 24 (2), 43–99 (1969)].
I. V. Astashova, “On Uniform Estimates of Positive Solutions of Quasilinear Differential Equations,” Diff. Uravn. 41(11), 1579–1580 (2005) [Diff. Eqns. 41, 1656–1657 (2005)].
I. V. Astashova, “Uniform Estimates for Positive Solutions to Quasi-linear Differential Equations of Even Order,” Tr. Semin. im. I.G. Petrovskogo 25, 21–34 (2005) [J. Math. Sci. 135 (1), 2616–2624 (2006)].
E. Mitidieri and S. I. Pohozaev, A Priori Estimates and Blow-up of Solutions to Nonlinear Partial Differential Equations and Inequalities (Nauka, Moscow, 2001), Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 234 [Proc. Steklov Inst. Math. 234 (2001)].
J. Hay, “Necessary Conditions for the Existence of Global Solutions of Higher-Order Nonlinear Ordinary Differential Inequalities,” Diff. Uravn. 38(3), 344–350 (2002) [Diff. Eqns. 38, 362–368 (2002)].
A. A. Kon’kov, “On Solutions of Non-autonomous Ordinary Differential Equations,” Izv. Ross. Akad. Nauk, Ser. Mat. 65(2), 81–126 (2001) [Izv. Math. 65, 285–327 (2001)].
I. V. Astashova, “Uniform Estimates for Positive Solutions of Quasilinear Differential Equations,” Dokl. Akad. Nauk 409(5), 586–590 (2006) [Dokl. Math. 74 (1), 555–558 (2006)].
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © I.V. Astashova, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 261, pp. 26–36.
Rights and permissions
About this article
Cite this article
Astashova, I.V. Uniform estimates for positive solutions of higher order quasilinear differential equations. Proc. Steklov Inst. Math. 261, 22–33 (2008). https://doi.org/10.1134/S008154380802003X
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S008154380802003X