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Dirichlet problems for linear and semilinear sub-Laplace equations on Carnot groups

Zixia Yuan1* and Guanxiu Yuan2

Author Affiliations

1 School of Mathematical Science, University of Electronic Science and Technology of China, Chengdu 611731, China

2 Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China

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Journal of Inequalities and Applications 2012, 2012:136  doi:10.1186/1029-242X-2012-136

Published: 12 June 2012


The purpose of this article, is to study the Dirichlet problems of the sub-Laplace equation Lu + f(ΞΎ, u) = 0, where L is the sub-Laplacian on the Carnot group G and f is a smooth function. By extending the Perron method in the Euclidean space to the Carnot group and constructing barrier functions, we establish the existence and uniqueness of solutions for the linear Dirichlet problems under certain conditions on the domains. Furthermore, the solvability of semilinear Dirichlet problems is proved via the previous results and the monotone iteration scheme corresponding to the sub-Laplacian.

Mathematics Subject Classifications: 35J25, 35J70, 35J60.

Carnot group; sub-Laplace equation; Dirichlet problem; Perron method; monotone iteration scheme