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

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.

DOI: https://doi.org/10.4213/im313

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

Bibliographic databases:

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

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
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. G. G. Laptev, “Non-existence of solutions of semilinear parabolic differential inequalities in cones”, Sb. Math., 192:10 (2001), 1471–1490
3. G. G. Laptev, “Nonexistence of Solutions of Elliptic Differential Inequalities in Conic Domains”, Math. Notes, 71:6 (2002), 782–793
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
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
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
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
8. G. G. Laptev, “Absence of solutions to higher-order evolution differential inequalities”, Siberian Math. J., 44:1 (2003), 117–131
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
10. Laptev G., “Nonexistence Results for Higher-Order Evolution Partial Differential Inequalities”, Proc. Amer. Math. Soc., 131:2 (2003), 415–423
11. A. A. Kon'kov, “Behavior of Solutions of Quasilinear Elliptic Inequalities”, Journal of Mathematical Sciences, 134:3 (2006), 2073–2237
12. Kon'kov A., “Comparison Theorems for Second-Order Elliptic Inequalities”, Nonlinear Anal.-Theory Methods Appl., 59:4 (2004), 583–608
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
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
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
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
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