Subelliptic equations with singular nonlinearities on the Heisenberg group
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 th...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2018-01-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13661-018-0925-y |
| Summary: | 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. |
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| ISSN: | 1687-2770 |