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Báo cáo hóa học: "Research Article A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem"

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem | Hindawi Publishing Corporation Journal ofInequalities and Applications Volume 2007 Article ID 32585 24 pages doi 10.1155 2007 32585 Research Article A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem Jingbo Dou Pengcheng Niu and Zixia Yuan Received 22 March 2007 Revised 26 May 2007 Accepted 20 October 2007 Recommended by László Losonczi Based on properties of vector fields we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the p-sub-Laplacian. The asymptotic behaviour simplicity and isolation of the first eigenvalue are also considered. Copyright 2007 Jingbo Dou 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. 1. Introduction Let Lpựu -kH pU - ựVp udp u 0 ự Q - 1.1 be the Hardy operator on the Heisenberg group. We consider the following weighted eigenvalue problem with a singular weight Lpựu Ằf u p-2u in o c H 1.2 u 0 on do where 1 p Q 2 2 A e R f e p f O R limd 0 dp f Ỉ yp 0 f e Ll c o 0 o is abounded domain in the Heisenberg group and the definitions of d and yp see below. We investigate the weak solution of 1.2 and the asymptotic behavior of the first eigenvalue for different singular weights as ự increases to Q - p p p. Furthermore we show that the first eigenvalue is simple and isolated as 2 Journal of Inequalities and Applications well as the eigenfunctions corresponding to other eigenvalues change sign. Our proof is mainly based on a Hardy inequality with remainder terms. It is established by the vector field method and an elementary integral inequality. In addition we show that the constants .