Abstract
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:
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.
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This work was supported by the National Natural Science Foundation of China (Grant Nos. 10871157 and 11001221), Specialized Research Fund for the Doctoral Program of Higher Education (No. 200806990032) and Northwestern Polytechnical University Jichu Yanjiu Jijin Tansuo Xiangmu (No. GBKY 1020).
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Feng, X., Niu, P. Regularity lifting of weak solutions for nonlinear sub-Laplace equations on homogeneous groups. Arch. Math. 98, 361–371 (2012). https://doi.org/10.1007/s00013-012-0377-z
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DOI: https://doi.org/10.1007/s00013-012-0377-z