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The Behavior of Solutions to the Dirichlet Problem for Second Order Elliptic Equations with Variable Nonlinearity Exponent in a Neighborhood of a Conical Boundary Point

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We study the Dirichlet problem for the p-Laplacian in a conical domain with the homogeneous boundary condition on the lateral surface of a cone with vertex at the origin. We assume that the variable exponent p = p(x) is separated from 1 and ∞ and denote by Ω the intersection of the cone with the unit (n − 1)-dimensional sphere. We prove that (i) if p satisfies the Lipschitz condition and ∂Ω is of class C 2+β, then the solution to the Dirichlet problem is O(|x|λ) in a neighborhood of the origin, where λ is the sharp exponent of tending to zero of solutions to the same Dirichlet problem for the p(0)-Laplacian and (ii) if p satisfies the Hölder condition, p(0) = 2, and ∂Ω is of class C 1+β, then the solution to the Dirichlet problem is O(|x|λ0) in a neighborhood of the origin, where λ 0 is the sharp exponent of tending to zero of solutions to the same Dirichlet problem for the Laplace operator. Bibliography: 18 titles.

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Authors and Affiliations

  1. A. G. and N. G. Stoletov Vladimir State University, 87, Gor’kogo St., Vladimir, 600000, Russia

    Yu. Alkhutov

  2. University of Warmia and Mazury, 2, Michala Oczapowskiego St., Olsztyn, Poland

    M. V. Borsuk

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  1. Yu. Alkhutov
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  2. M. V. Borsuk
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Correspondence to Yu. Alkhutov.

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Translated from Problemy Matematicheskogo Analiza 81, August 2015, pp. 3–28.

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Alkhutov, Y., Borsuk, M.V. The Behavior of Solutions to the Dirichlet Problem for Second Order Elliptic Equations with Variable Nonlinearity Exponent in a Neighborhood of a Conical Boundary Point. J Math Sci 210, 341–370 (2015). https://doi.org/10.1007/s10958-015-2570-7

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