Geodesic
Elliptic and parabolic inequalities with point singularities on the boundary
Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1417-1437 Cet article a éte moissonné depuis la source Math-Net.Ru

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It is shown that various quasilinear elliptic and parabolic differential inequalities and systems of such inequalities defined on bounded domains, and which have point singularities on the boundary do not have solutions. The method of nonlinear capacity is used in the proof. Examples show that the conditions obtained by this method cannot be improved in the class of problems under consideration. Bibliography: 14 titles.
Keywords: quasilinear equations, nonexistence of solutions, boundary singularities. 
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E. I. Galakhov. Elliptic and parabolic inequalities with point singularities on the boundary. Sbornik. Mathematics, Tome 200 (2009) no. 10, pp. 1417-1437. http://geodesic.mathdoc.fr/item/SM_2009_200_10_a0/

[1] I. T. Kiguradze, T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations, Math. Appl. (Soviet Ser.), 89, Kluwer, Dordrecht, 1993 | MR | Zbl | Zbl

[2] J. Hay, “Necessary conditions for the existence of local solutions to higher-order singular nonlinear ordinary differential equations and inequalities”, Dokl. Math., 67:1 (2003), 66–69 | MR | Zbl

[3] J. Hay, “On necessary conditions for the existence of local solutions to singular nonlinear ordinary differential equations and inequalities”, Math. Notes, 72:5–6 (2002), 847–857 | DOI | MR | Zbl

[4] H. Breźis, X. Cabré, “Some simple nonlinear PDE's without solutions”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1:2 (1998), 223–262 | MR | Zbl

[5] A. V. Demyanov, A. I. Nazarov, “On solvability of the Dirichlet problem to the semilinear Schrödinger equation with singular potential”, J. Math. Sci. (N. Y.), 143:2 (2007), 2857–2868 | DOI | MR | Zbl

[6] I. Kombe, “Doubly nonlinear parabolic equations with singular lower order term”, Nonlinear Anal., 56:2 (2004), 185–199 | DOI | MR | Zbl

[7] E. Mitidieri, S. I. Pokhozhaev, “The absence of global positive solutions of quasilinear elliptic inequalities”, Dokl. Math., 57:2 (1998), 250–253 | MR | Zbl

[8] E. Mitidieri, S. I. Pokhozhaev, “Nonexistence of positive solutions for a system of quasilinear elliptic equations and inequalities in $\mathbb{R}^N$”, Dokl. Math., 59:3 (1999), 351–355 | MR | Zbl

[9] E. Mitidieri, S. I. Pokhozhaev, “Nonexistence of positive solutions for quasilinear elliptic problems in $\mathbb{R}^N$”, Proc. Steklov Inst. Math., 227:4 (1999), 186–216 | MR | Zbl

[10] E. Mitidieri, S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities”, Proc. Steklov Inst. Math., 234:3 (2001), 1–362 | MR | Zbl | Zbl

[11] M.-F. Bidaut-Véron, S. Pohozaev, “Nonexistence results and estimates for some nonlinear elliptic problems”, J. Anal. Math., 84:1 (2001), 1–49 | DOI | MR | Zbl

[12] E. I. Galakhov, “On differential inequalities with point singularities on the boundary”, Proc. Steklov Inst. Math., 260:1 (2008), 112–122 | DOI | MR

[13] J. Serrin, “Local behavior of solutions of quasi-linear equations”, Acta Math., 111:1 (1964), 247–302 | DOI | MR | Zbl

[14] J. L. Vázquez, “A strong maximum principle for some quasilinear elliptic equations”, Appl. Math. Optim., 12:1 (1984), 191–202 | DOI | MR | Zbl