Dirichlet problems for linear and semilinear sub-Laplace equations on Carnot groups
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
Journal of Inequalities and Applications 2012, 2012:136 doi:10.1186/1029-242X-2012-136Published: 12 June 2012
The purpose of this article, is to study the Dirichlet problems of the sub-Laplace equation Lu + f(ξ, 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.