Article

 Title: Positive solutions of inequality with $p$-Laplacian in exterior domains (English) Author: Mařík, Robert Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 Volume: 127 Issue: 4 Year: 2002 Pages: 597-604 Summary lang: English . Category: math . Summary: In the paper the differential inequality $\Delta _p u+B(x,u)\le 0,$ where $\Delta _p u=\div (\Vert \nabla u\Vert ^{p-2}\nabla u)$, $p>1$, $B(x,u)\in C(\mathbb{R}^{n}\times \mathbb{R},\mathbb{R})$ is studied. Sufficient conditions on the function $B(x,u)$ are established, which guarantee nonexistence of an eventually positive solution. The generalized Riccati transformation is the main tool. (English) Keyword: $p$-Laplacian Keyword: oscillation criteria MSC: 35B05 MSC: 35J60 MSC: 35R45 idZBL: Zbl 1074.35505 idMR: MR1942645 . Date available: 2009-09-24T22:05:42Z Last updated: 2012-06-18 Stable URL: http://hdl.handle.net/10338.dmlcz/133960 . Reference: [1] J. I. Díaz: Nonlinear Partial Differential Equations and Free Boundaries.Vol. I, Elliptic Equations, Pitman Publ., London, 1985. MR 0853732 Reference: [2] O. Došlý, R. Mařík: Nonexistence of the positive solutions of partial differential equations with $p$-Laplacian.Acta Math. Hungar. 90 (2001), 89–107. MR 1910321 Reference: [3] J. Jaroš, T. Kusano, N. Yoshida: A Picone type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equation of second order.Nonlinear Anal. Theory Methods Appl. 40 (2000), 381–395. MR 1768900 Reference: [4] R. Mařík: Hartman-Wintner type theorem for PDE with $p$-Laplacian.EJQTDE, Proc. 6th Coll. QTDE, 2000, No. 18, 1–7. MR 1798668 Reference: [5] R. Mařík: Oscillation criteria for PDE with $p$-Laplacian via the Riccati technique.J. Math. Anal. Appl. 248 (2000), 290–308. MR 1772598 Reference: [6] E. W. Noussair, C. A. Swanson: Oscillation of semilinear elliptic inequalities by Riccati equation.Can. J. Math. 22 (1980), 908–923. Reference: [7] C. A. Swanson: Semilinear second order ellitpic oscillation.Can. Math. Bull. 22 (1979), 139–157. MR 0537295 .

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