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Izv. RAN. Ser. Mat., 2000, Volume 64, Issue 6, Pages 107–124 (Mi izv313)  

This article is cited in 16 scientific papers (total in 16 papers)

The absence of global positive solutions of systems of semilinear elliptic inequalities in cones

G. G. Laptev

Abstract: Let $K$ be a cone in $\mathbb R^N$, $N\geqslant 2$. We establish conditions for the absence of global non-trivial non-negative solutions of semilinear elliptic inequalities and systems of inequalities of the form
$$ -\operatorname{div}(|x|^\alpha Du)\geqslant |x|^\beta u^q, \qquad u|_{\partial K}=0. $$
We find the critical exponent $q^*$ that divides the domains of existence of these solutions from those of their absence. We prove that in the limiting case $q=q^*$ there are no solutions. The method is to multiply the system by a special factor and integrate the inequalities thus obtained.


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English version:
Izvestiya: Mathematics, 2000, 64:6, 1197–1215

Bibliographic databases:

MSC: 35J60, 35G20, 35B99, 35J65, 35B50
Received: 18.05.1999

Citation: G. G. Laptev, “The absence of global positive solutions of systems of semilinear elliptic inequalities in cones”, Izv. RAN. Ser. Mat., 64:6 (2000), 107–124; Izv. Math., 64:6 (2000), 1197–1215

Citation in format AMSBIB
\by G.~G.~Laptev
\paper The absence of global positive solutions of systems of semilinear elliptic inequalities in cones
\jour Izv. RAN. Ser. Mat.
\yr 2000
\vol 64
\issue 6
\pages 107--124
\jour Izv. Math.
\yr 2000
\vol 64
\issue 6
\pages 1197--1215

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    This publication is cited in the following articles:
    1. E. Mitidieri, S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities”, Proc. Steklov Inst. Math., 234 (2001), 1–362  mathnet  mathscinet  zmath
    2. G. G. Laptev, “Non-existence of solutions of semilinear parabolic differential inequalities in cones”, Sb. Math., 192:10 (2001), 1471–1490  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    3. G. G. Laptev, “Nonexistence of Solutions of Elliptic Differential Inequalities in Conic Domains”, Math. Notes, 71:6 (2002), 782–793  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. J. Hay, “On Necessary Conditions for the Existence of Local Solutions to Singular Nonlinear Ordinary Differential Equations and Inequalities”, Math. Notes, 72:6 (2002), 847–857  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. G. G. Laptev, “Absence of solutions of differential inequalities and systems of hyperbolic type in conic domains”, Izv. Math., 66:6 (2002), 1147–1170  mathnet  crossref  crossref  mathscinet  zmath
    6. G. G. Laptev, “On the nonexistence of solutions of elliptic differential inequalities in a neighborhood of a conic point of the boundary”, Russian Math. (Iz. VUZ), 46:9 (2002), 48–57  mathnet  mathscinet  zmath  elib
    7. Hay J., “Necessary Conditions for the Existence of Global Solutions of Higher-Order Nonlinear Ordinary Differential Inequalities”, Differ. Equ., 38:3 (2002), 362–368  mathnet  crossref  mathscinet  zmath  isi  scopus
    8. G. G. Laptev, “Absence of solutions to higher-order evolution differential inequalities”, Siberian Math. J., 44:1 (2003), 117–131  mathnet  crossref  mathscinet  zmath  isi
    9. G. G. Laptev, “Non-existence of global solutions for higher-order evolution inequalities in unbounded cone-like domains”, Mosc. Math. J., 3:1 (2003), 63–84  mathnet  mathscinet  zmath
    10. Laptev G., “Nonexistence Results for Higher-Order Evolution Partial Differential Inequalities”, Proc. Amer. Math. Soc., 131:2 (2003), 415–423  crossref  mathscinet  zmath  isi  scopus
    11. A. A. Kon'kov, “Behavior of Solutions of Quasilinear Elliptic Inequalities”, Journal of Mathematical Sciences, 134:3 (2006), 2073–2237  mathnet  crossref  mathscinet  zmath  elib
    12. Kon'kov A., “Comparison Theorems for Second-Order Elliptic Inequalities”, Nonlinear Anal.-Theory Methods Appl., 59:4 (2004), 583–608  crossref  mathscinet  zmath  isi  scopus
    13. de Figueiredo D.G., Sirakov B., “Liouville type theorems, monotonicity results and a priori bounds for positive solutions of elliptic systems”, Math. Ann., 333:2 (2005), 231–260  crossref  mathscinet  zmath  isi  scopus
    14. Kondratiev V., Liskevich V., Moroz V., “Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains”, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22:1 (2005), 25–43  crossref  mathscinet  zmath  adsnasa  isi  scopus
    15. Dancer E.N., Wei J., Weth T., “A Priori Bounds Versus Multiple Existence of Positive Solutions for a Nonlinear Schrodinger System”, Ann. Inst. Henri Poincare-Anal. Non Lineaire, 27:3 (2010), 953–969  crossref  mathscinet  zmath  adsnasa  isi  scopus
    16. Armstrong S.N., Sirakov B., “Nonexistence of Positive Supersolutions of Elliptic Equations via the Maximum Principle”, Commun. Partial Differ. Equ., 36:11 (2011), 2011–2047  crossref  mathscinet  zmath  isi  elib  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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