Abstract
The work deals with the Dirichlet problem for elliptic equations with nonhomogeneous anisotropic degeneracy in a possibly unbounded domain of multidimensional Euclidean space. The existence of weak solutions is proved. Some conditions are established connecting the character of nonlinearity of the equation and the geometric characteristics of the domain which guarantee the one-dimensional localization (vanishing) of weak solutions. The equation with anisotropic degeneracy is shown to admit localized solutions even in the absence of absorption.
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Dedicated to the memory of Tadei Ivanovich Zelenyak, an outstanding mathematician.
Original Russian Text Copyright © 2005 Antontsev S. N. and Shmarev S. I.
The first author was supported by the Mathematical Center of the University of Beira Interior (Portugal); the second, by the research grants MTM-2004-05417 (Spain) and HPRN-CT-2002-00274 (EC).
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Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 5, pp. 963–984, September–October, 2005.
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Antontsev, S.N., Shmarev, S.I. On Localization of Solutions of Elliptic Equations with Nonhomogeneous Anisotropic Degeneracy. Sib Math J 46, 765–782 (2005). https://doi.org/10.1007/s11202-005-0076-0
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DOI: https://doi.org/10.1007/s11202-005-0076-0