Abstract
We prove nonexistence theorems of nonnegative nontrivial solutions for a system of ordinary differential inequalities in bounded domains with singular points on the boundary. The proofs are based on the test function using a method developed by Mitidieri and Pohozaev. We also give the examples demonstrating that the conditions obtained are sharp in the case of the problem under consideration.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. T. Kiguradze and T. A. Chanturia, “Asymptotic properties of solutions of nonautonomous ordinary differential equations,” Mathematics and Its Applications 89 (1993).
J. Hay, “On necessary conditions for the existence of local solutions to singular ordinary differential equations and inequalties,” Math. Notes 72 (5), 847–857 (2002).
J. Hay, “Necessary conditions for the existence of global solutions of higher-order nonlinear ordinary differential inequalities,” Differential Equations 38 (3), 362–368 (2002).
J. Hay, “Necessary conditions for the existence of local solutions to higher-order singular nonlinear ordinary differential equations and inequalities,” Dokl. Math. 67 (1), 66–69 (2003).
L. D’Ambrosio, “Liouville theorems for anisotropic quasilinear inequalities,” Nonlinear Anal. 70, 2855–2869 (2009).
R. Filippucci, “Nonexistence of positive weak solutions of elliptic inequalities,” Nonlinear Anal. 70, 2903–2916 (2009).
R. Filippucci, “Nonexistence of nonnegative solutions of elliptic systems of divergence type,” J. Differential Equations 250, 572–595 (2011).
E. Galakhov, “Some nonexistence results for quasilinear elliptic problems,” Math. Anal. Appl. 252, 256–277 (2000).
E. Galakhov, “Positive solutions of some semilinear differential inequalities and systems,” Differ. Equ. 40, 711–722 (2004).
E. Galakhov, “Positive solutions of some quasilinear partial differential inequalities and systems,” Math. Nachr. 279, 831–842 (2006).
E. Galakhov, “Some nonexistence results for quasilinear PDE’s,” Commun. Pure Appl. Anal. 6, 141–161 (2007).
E. Galakhov, “On differential inequalities with point singularities on the boundary,” Proc. Steklov Inst. Math. 260 (1), 112–122 (2008).
G. G. Laptev, “On the absence of solutions to a class of singular semilinear differential inequalities,” Proc. Steklov Inst. Math. 232, 216–228 (2001).
X. Li and F. Li, “A priori estimates for nonlinear differential inequalities and applications,” J. Math. Anal. Appl. 378 (2), 723–733 (2011).
X. Li and F. Li, “Nonexistence of solutions for nonlinear differential inequalities with gradient nonlinearities,” Commun. Pure Appl. Anal. 11 (3), 935–943 (2012).
X. Li and F. Li, “Nonexistence of solutions for singular quasilinear differential inequalities with a gradient nonlinearity,” Nonlinear Anal. 75 (5), 2812–2822 (2012).
E. Mitidieri and S. I. Pohozaev, “A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities,” Steklov Inst. Math. 234, 1–362 (2001).
E. Mitidieri and S. I. Pohozaev, “The absence of global positive solutions to quasilinear elliptic inequalities,” Dokl. Akad. Nauk. 57, 250–253 (1998).
E. Mitidieri and S. I. Pohozaev, “Nonexistence of positive solutions for systems of quasilinear elliptic equations and inequalities in \( \mathbb{R}^N\),” Dokl. Akad. Nauk. 59, 351–355 (1999).
E. Mitidieri and S. I. Pohozaev, “Absence of positive solutions for quasilinear elliptic problems in \(\mathbb{R}^N\),” Steklov Inst. Math. 227, 186–216 (1999).
E. Mitidieri and S. I. Pohozaev, “Nonexistence of weak solutions for some degenerate elliptic and parabolic problems on \(\mathbb R^n\),” J. Evol. Equ. 1, 189–220 (2001).
E. Mitidieri and S. I. Pohozaev, “Towards a unified approach to nonexistence of solutions for a class of differential inequalities,” Milan J. Math. 72, 129–162 (2004).
S. I. Pohozaev, “A general approach to the theory of the nonexistence of global solutions of nonlinear partial differential equations and inequalities,” Steklov Inst. Math. 236, 273–284 (2002).
S. I. Pohozaev, “Nonexistence of local solutions to semilinear partial differential inequalities,” Ann. Inst. H. Poincaré Anal. Non Linéaire 21, 487–502 (2004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Li, X., Wan, H. & Li, X. Absence of Solutions for a System of Ordinary Differential Inequalities. Math Notes 109, 600–608 (2021). https://doi.org/10.1134/S0001434621030299
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434621030299