International Journal of Differential Equations
Volume 2011 (2011), Article ID 198606, 23 pages
http://dx.doi.org/10.1155/2011/198606
Research Article

On Carlson's Type Removability Test for the Degenerate Quasilinear Elliptic Equations

Farman I. Mamedov, Aslan D. Quliyev, and Mirfaig M. Mirheydarli

Institute of Mathematics and Mechanics, National Academy of Sciences, Baku Az 141, F. Agaev 9, Azerbaijan

Received 27 May 2011; Accepted 13 August 2011

Academic Editor: Xingfu Zou

Copyright © 2011 Farman I. Mamedov et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Carlson's type theorem on removable sets for 𝛼 -Holder continuous solutions is investigated for the quasilinear elliptic equations d i v 𝐴 ( 𝑥 , 𝑢 , ∇ 𝑢 ) = 0 , having degeneration 𝜔 in the Muckenhoupt class. In partial, when 𝛼 is sufficiently small and the operator is weighted 𝑝 -Laplacian, we show that the compact set 𝐸 is removable if and only if the Hausdorff measure Λ 𝜔 − 𝑝 + ( 𝑝 − 1 ) 𝛼 ( 𝐸 ) = 0 .