Abstract
Let Ω be an arbitrary open set in R n , and let σ(x) and g i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W r p (Ω, σ, \( \vec g \)), where r ∈ N, p ≥ 1, and \( \vec g \)(x) = (g 1(x), g 2(x), ..., g n (x)), with the norm
where
We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W r p (Ω, σ, \( \vec g \)) for the functional
where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.
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Original Russian Text © S.A. Iskhokov, 2008, published in Differentsial’nye Uravneniya, 2008, Vol. 44, No. 2, pp. 232–245.
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Iskhokov, S.A. On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration. Diff Equat 44, 241–255 (2008). https://doi.org/10.1134/S0012266108020122
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DOI: https://doi.org/10.1134/S0012266108020122