Subelliptic equations with singular nonlinearities on the Heisenberg group

Abstract

Abstract In this paper, we consider the Dirichlet boundary value problem to singular semilinear subelliptic equation on the Heisenberg group − Δ H u = 1 u γ + f ( u ) , γ > 0 . $$-\Delta_{\mathbb{H}}u=\frac{1}{u^{\gamma}}+f(u), \quad \gamma>0. $$ We prove the positivity and continuity up to the boundary for the weak solutions. We also conclude monotonicity of cylindrical solutions to the problem based on a study of the equation − Δ H u 0 = 1 u 0 γ $-\Delta_{\mathbb {H}}u_{0}=\frac{1}{u_{0}^{\gamma}}$ . The main technique is a generalization of the moving plane method to the Heisenberg group.

Keywords

• Heisenberg group
• subelliptic equation
• singular nonlinearity
• monotonicity
• moving plane method