Archiv der Mathematik

, Volume 98, Issue 4, pp 361–371 | Cite as

Regularity lifting of weak solutions for nonlinear sub-Laplace equations on homogeneous groups

  • Xiaojing Feng
  • Pengcheng Niu


Let G be a homogeneous group, and let X 1, X 2, … , X m be left invariant real vector fields being homogeneous of degree one on G. We consider the following Dirichlet boundary value problem of the sub-Laplace equation involving the critical exponent and singular term:
$$\left\{\begin{array}{ll}-\sum_{j=1}^{m}X_j^2u(x)-\frac{a}{\|x\|^\nu}u(x)=u^{\frac{Q+2}{Q-2}}(x), x\in\Omega,\\ u(x)=0, \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\, x\in \partial\Omega,\end{array}\right.$$
where \({\Omega\subset G}\) is a bounded domain with smooth boundary and \({\mathbf{0}\in\Omega}\) , Q is the homogeneous dimension of G, \({a\in \mathbb{R},\ \nu <2 }\) . We boost u to \({L^p(\Omega)}\) for any \({1\leq p < \infty}\) if \({u\in S^{1,2}_0(\Omega)}\) is a weak solution of the problem above.

Mathematics Subject Classification

Primary 35R03 35B33 Secondary 35J08 57R25 


Regularity lifting Homogeneous group Vector fields Bootstrap method 


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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Department of Applied Mathematics, Key Laboratory of Space Applied Physics and Chemistry, Ministry of EducationNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

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