Abstract
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.
Highlights
In this article we consider Dirichlet problems of the typeLu + f (ξ, u) = 0, in, u = φ, on ∂, (1:1)where Ω is a bounded domain in a Carnot group G and L is the sub-Laplacian
Where Ω is a bounded domain in a Carnot group G and L is the sub-Laplacian
Thanks to the previous results, Capogna et al [4] established the solvability of the Dirichlet problem when the boundary datum belongs to Lp, 1
Summary
Where Ω is a bounded domain in a Carnot group G and L is the sub-Laplacian. Some knowledge on G and L see section. We try to extend the existence of solutions for semilinear Dirichlet problems on the Heisenberg balls in [9] to general Carnot domains. Based on the work in [3], we construct a barrier function in a domain of the Carnot group (see Lemma 3.10) under the hypothesis of the outer sphere condition to discuss the boundary behaviour of the Perron solutions. By finding a barrier function related to the sub-Laplacian L, we prove that the Perron solutions for linear Dirichlet problems are continuous up to the boundary. In the space Sk, p, we shall adopt the norm f
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