Solution Of A Class Of Nonlinear Equations. On The Heisenberg Group Of Vmo

In this paper, we consider the homogeneous Dirichlet problem as follows:in an open bounded setΩof the Heisenberg group Hⁿ, whereμis a Radonmeasure, and a: Hⁿ×R²ⁿâ†'R²ⁿ is a Carathéodory function satisfying somecoerciveness and monotonicity assumptions. We transfer the measureμto aC^∞function via the standard mollification, so that we can use J.Leray andJ.L.Lions's existence theorem of the boundary problem of a nonlinear monotonicoperator to prove the existence of a solution which belongs to the Sobolev spaceS₀^1,q(Ω). Combing and the energy inequality and some prior estimates on Hisen-berg groups, we finally prove that such a problem admits a solution u which islocally BMO inΩ. Furthermore, the solution u is locally VMO ifμcontains noatoms.